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Example 19.5. VaR and ES for a position in an S&P 500 index fund using a GARCH(1,1) model | An AR(1)+GARCH(1,1) model was fit to the log returns on the S&P 500. The \( \mathrm{{AR}}\left( 1\right) \) coefficient was small and not significantly different from 0, so a \( \mathrm{{GARCH}}\left( {1,1}\right) \) was used for estimation of \( \mathrm{{VaR}} \) and \( \mathrm{{ES}} \) . The \( \mathrm{{GARCH}}\left(... | No |
Example 19.6. VaR and ES for portfolios of the three stocks in the CRSPday data set | This example uses the data set CRSPday used earlier in Examples 7.1 and 7.4. There are four variables - returns on GE, IBM, Mobil, and the CRSP index and we found in Example 7.4 that their returns can be modeled as having a multivariate \( t \) -distribution with tail index 5.94 . In this example, we will only consider... | Yes |
For an investment of $4,000, what are estimates of \( {\operatorname{VaR}}^{t}\left( {0.05}\right) \) and \( E{S}^{t}\left( {0.05}\right) \) ? | Now, fit a ARMA(0,0)+GARCH(1,1) model to the returns and calculate one step forecasts.\n\nlibrary(fGarch) # for qstd() function\n\nlibrary(rugarch)\n\ngarch.t \( = \) ugarchspec(mean.model=list(armaOrder=c \( \left( {0,0}\right) \) ),\n\nvariance.model=list(garchOrder=c(1,1)),\n\ndistribution.model \( = \) \ | No |
Updating the prior beliefs about the probability that a stock price will increase | We continue Example 20.1 but change the simple, but unrealistic, prior that said that \( \theta \) was either 0.4 or 0.6 to a more plausible prior where \( \theta \) could be any value in the interval \( \left\lbrack {0,1}\right\rbrack \), but with values near \( 1/2 \) more likely. Specifically, we use a \( \operatorn... | Yes |
Computing the posterior density of the probability that a stock price will increase-general case of a conjugate prior | Suppose now that the prior for \( \theta \) is \( \operatorname{Beta}\left( {\alpha ,\beta }\right) \) so that the prior density is\n\n\[ \pi \left( \theta \right) = {K}_{1}{\theta }^{\alpha - 1}{\left( 1 - \theta \right) }^{\beta - 1}, \]\n\nwhere \( {K}_{1} \) is a constant. Suppose that the stock price is observed o... | Yes |
Now suppose that \( {Y}_{1},\ldots ,{Y}_{n} \) are i.i.d. with a known mean \( \mu \) and an unknown variance \( {\sigma }^{2} \) and precision \( \tau = 1/{\sigma }^{2} \) . We will show that the conjugate priors for \( \tau \) are the gamma distributions and we will find the posterior distribution of \( \tau \) . Def... | Simple algebra shows that the likelihood is\n\n\[ \n f\left( {{Y}_{1},\ldots ,{Y}_{n} \mid \tau }\right) \propto \exp \left( {-\frac{1}{2}{n\tau }{s}^{2}}\right) {\tau }^{n/2}. \]\n\n(20.26)\n\nLet the prior distribution be the gamma distribution with shape parameter \( \alpha \) and scale parameter \( b \) which has d... | Yes |
Posterior interval for a normal mean when the variance is known | This example continues Example 20.4. By (20.20) and (20.21), a \( \left( {1 - \alpha }\right) \) \( {100}\% \) posterior interval for \( \mu \) is\n\n\[ \frac{{\tau }_{\bar{Y}}\bar{Y} + {\tau }_{0}{\mu }_{0}}{{\tau }_{\bar{Y}} + {\tau }_{0}} \pm {z}_{\alpha /2}\sqrt{\frac{1}{\frac{n}{{\sigma }^{2}} + \frac{1}{{\sigma }... | Yes |
Using JAGS to fit a t-distribution to returns | In this example, a \( t \) -distribution will be fit to S&P 500 returns using JAGS called from R. The BUGS program below is in the file univt. bug. The program will run under any of OpenBUGS, WinBUGS, or JAGS. In this example, JAGS will be used.\n\nmodel\{\n\nfor(i in \( 1 : \mathrm{N} \) )\n\n\{\n\n\( r\left\lbrack i\... | Yes |
Example 20.11. \( {p}_{D} \) when estimating a normal mean with known precision | Suppose that \( \mathbf{Y} = \left( {{Y}_{1},\ldots ,{Y}_{n}}\right) \) are i.i.d. \( N\left( {\mu ,1}\right) \), so \( \mathbf{\theta } = \mu \) in this example. Then the log-likelihood is\n\n\[ \log \{ L\left( \mu \right) \} = - \frac{1}{2}\mathop{\sum }\limits_{{i = 1}}^{n}{\left( {Y}_{i} - \mu \right) }^{2} - \frac... | Yes |
Estimating the correlation matrix of the CRSPday data | In Example 7.4, the correlation matrix of the CRSPday returns data was estimated by maximum likelihood. In this example, the MLE will be compared to a Bayes estimate and the two estimates will be found to be very similar. The BUGS program used in this example is\n\nmodel{\n\nfor \( \left( {i\text{in}1 : N}\right) \)\n\... | No |
Fitting an ARMA(1,1) stochastic volatility model to the S&P 500 stock returns | As an illustration, model (20.61)-(20.63) will be fit to daily S&P 500 log returns from January 2011 through October 2014. Model (20.61) will be used with no covariates so that \( {Y}_{t} = \mu + {a}_{t} \) . An \( \operatorname{ARMA}\left( {1,1}\right) \) stochastic volatility model will be used so that \( \log \left(... | No |
Fitting a GARCH(1,1) model to the S&P 500 stock returns | The following \( R \) code fits a \( \operatorname{GARCH}\left( {1,1}\right) \) model to the data in Example 20.14.\n\nlibrary(rjags)\n\ndat \( = \) read.csv(\ | No |
Fitting a one factor model to stock returns | In this example, the factor will be the returns on the S&P 500 so this is a Bayesian version of the CAPM. The model that will be used is\n\n\[ \n{R}_{j, t} = {\beta }_{j}{R}_{M, t} + {\epsilon }_{j, t} \n\]\n\nwhere, as in Chap. \( {17},{R}_{j, t} \) is the return on the \( j \) stock at time \( t, j = 1,\ldots ,{10} \... | Yes |
Problem 2\n\n(a) Which parameter mixes best and which mixes worse according to the ACF plots? Explain your answers.\n\n(b) Find the posterior skewness and kurtosis of the degrees of freedom parameter. | The function densityplot () gives kernel density estimate from each chain.\n\nlibrary(lattice)\n\ndensityplot(univ_t.coda) | No |
Problem 3 Which posterior densities are most skewed? | The kurtosis of a \( t \) -distribution is \( 3\left( {\nu - 2}\right) /\left( {\nu - 4}\right) \) if \( \nu > 4 \) and is \( + \infty \) if \( \nu \leq 4 \) . Variables in \( \mathrm{R} \) can have infinite values: Inf is \( + \infty \) and \( - \operatorname{Inf} \) is \( - \infty \), so \( \mathrm{R} \) can handle i... | Yes |
Local polynomial estimation of forward rates. | The \( \mathrm{R} \) code in this section computes two estimates of the forward rate function, a parametric estimate based on the Nelson-Siegel model and a nonparametric estimates using local polynomial smoothing. The program is split into several chucks of code with discussion between the chucks.\n\nLine 1 reads in pr... | Yes |
Example 21.2. Estimating the drift and volatility for the evolution of the risk-free returns | In this example, we return to estimating the drift and squared volatility functions for the evolution of the risk-free returns. Three estimators will be used: local linear, local quadratic, and a penalized spline. The R code is:\n\nlibrary(Ecdat)\nlibrary(KernSmooth)\nlibrary(locfit)\nlibrary(mgcv)\n\ndata(Capm)\n\n\\(... | Yes |
Consider the observed time series, of length \( n = {25} \), reported in the last column of Tables A.2(a) and A.2(b) of the online supplementary material. Suppose we are interested in solving the single-lag testing problems for lags \( l = 2 \) and \( l = 3 \) . | If \( l = 3 \), the \( {n}_{3} = {22} \) pairs to be considered are highlighted in Table A.2(a). To classify these pairs in the \( k \times k \) contingency table reported in Table 1, the value of \( k \) first has to be determined. Based on (6) the value \( k = 2 \) is obtained, since, in this case, \( {k}_{s} = 2 \) ... | Yes |
As an example of a situation of Case 2, consider the strictly stationary process\n\n\[ \n{X}_{t} = \operatorname{sign}\left( {{\varepsilon }_{t - 1}{\varepsilon }_{t - 2}}\right) + {\varepsilon }_{t} = \left\{ {\begin{array}{ll} 1 + {\varepsilon }_{t} & \text{ if }{\varepsilon }_{t - 1}{\varepsilon }_{t - 2} \geq 0 \\ ... | In Appendix B of the online supplementary material, it is proved that \( {\left\{ {X}_{t}\right\} }_{t \in \mathbb{N}} \) is a dependent process with the following features: (i) the random variables in \( \left( {{X}_{t - l},{X}_{t}}\right) \) are independent for any \( l \) ; (ii) the random variables in \( \left( {{X... | No |
By considering the data already presented in Example 2, we show how the multiple-lag ADF is able to detect the presence of serial dependence structures that are not captured by the Portmanteau ACF and by the Portmanteau ADF. | Figure 3 displays the rejection rates related to the test based on \( {\widehat{\delta }}_{L}, L \in \mathcal{P}\left( \mathcal{L}\right), L \neq \varnothing \), and \( \mathcal{L} = \) \( \{ 1,2,3,4,5\} \) . Similarly to Figure 1, which reports the results of the tests based on \( {\widehat{Q}}_{L} \) and \( {\widehat... | Yes |
Let \( {X}_{t} = \mathop{\sum }\limits_{{i = 1}}^{t}{\eta }_{i} \), where \( \left\{ {\eta }_{i}\right\} \) is i.i.d. with mean 0 and finite variance \( {\sigma }^{2} \), and \( t \in \mathbb{N} \) . Then its ACVF and ACF are | \[ {\gamma }_{t, s} = \operatorname{cov}\left( {{X}_{t},{X}_{s}}\right) = \left( {t \land s}\right) {\sigma }^{2};\;{\rho }_{t, s} = \frac{t \land s}{\sqrt{ts}},\;s, t \in \mathbb{N}, \] | Yes |
Proposition 1. A set is a neighbourhood of each of its points if and only if it is open. | The everyday sense of the word \ | No |
Proposition 2. If to each element \( x \) of a set \( \mathrm{X} \) there corresponds a set \( \mathfrak{B}\left( x\right) \) of subsets of \( \mathrm{X} \) such that the properties \( \left( {\mathrm{V}}_{\mathrm{I}}\right) ,\left( {\mathrm{V}}_{\mathrm{{II}}}\right) ,\left( {\mathrm{V}}_{\mathrm{{III}}}\right) \) and... | By Proposition \( r \), if there is a topology on \( X \) satisfying these conditions, the set of open sets for this topology is necessarily the set \( \mathfrak{O} \) of subsets A of \( \mathrm{X} \) such that for each \( x \in \mathrm{A} \) we have \( \mathrm{A} \in \mathfrak{B}\left( x\right) \) ; hence the uniquene... | Yes |
Proposition 3. If \( \mathrm{X} \) is a topological space, then for a set \( \mathfrak{B} \) of open subsets of \( \mathrm{X} \) to be a base of the topology of \( \mathrm{X} \) it is necessary and sufficient that for each \( x \in \mathrm{X} \) the set of \( \mathrm{V} \in \mathfrak{B} \) such that \( x \in \mathrm{V}... | It is clear that the condition is necessary. Conversely, if it is satisfied, then given any open set \( \mathrm{U} \) and any \( x \in \mathrm{U} \) there is an open set \( {\mathrm{V}}_{x} \in \mathfrak{B} \) such that \( x \in {\mathrm{V}}_{x} \subset \mathrm{U} \) . The union of the sets \( {\mathrm{V}}_{x} \) for \... | Yes |
Proposition 4. The union of a locally finite family of closed subsets of a topological space \( \mathrm{X} \) is closed in \( \mathrm{X} \) . | Let \( {\left( {\mathrm{F}}_{t}\right) }_{t \in \mathbf{I}} \) be a locally finite family of closed subsets of \( \mathrm{X} \), and suppose that \( x \in \mathrm{X} \) does not belong to \( \mathrm{F} = \mathop{\bigcup }\limits_{{i \in \mathrm{I}}}{\mathrm{F}}_{\mathrm{t}} \) ; then \( x \) has a neighbourhood V which... | Yes |
Proposition 5. Let \( \mathrm{A} \) be an open set in \( \mathrm{X} \) ; then for every subset \( \mathrm{B} \) of \( \mathrm{X} \) we have\n\n\[ A \cap \bar{B} \subset \overline{A \cap B}\text{. } \] | For suppose \( x \in \mathrm{A} \cap \overline{\mathrm{B}} \) ; then if \( \mathrm{V} \) is any neighbourhood of \( x,\mathrm{\;V} \cap \mathrm{A} \) is a neighbourhood of \( x \), since \( \mathrm{A} \) is open; hence \( \mathrm{V} \cap \mathrm{A} \cap \mathrm{B} \) is not empty and therefore \( x \) lies in the closu... | Yes |
Proposition 6. If \( \mathfrak{B} \) is a base of the topology of a topological space \( \mathrm{X} \), there is a dense set \( \mathrm{D} \) in \( \mathrm{X} \) such that \( \operatorname{Card}\left( \mathrm{D}\right) \leq \operatorname{Card}\left( \mathfrak{B}\right) \) . | We may restrict ourselves to the case in which none of the sets of \( \mathfrak{B} \) is empty (the non-empty sets of \( \mathfrak{B} \) already form a base of the topology of X). For each \( \mathrm{U} \in \mathfrak{B} \), let \( {x}_{\mathrm{v}} \) be a point of \( \mathrm{U} \) ; it follows from Proposition 3 of no.... | Yes |
Proposition 1. Let \( f \) be a mapping of a topological space \( \mathbf{X} \) into a topological space \( {\mathrm{X}}^{\prime } \). If \( f \) is continuous at \( x \), and if \( x \) lies in the closure of a subset \( \mathrm{A} \) of \( \mathbf{X} \), then \( f\left( x\right) \) lies in the closure of \( f\left( \... | Let \( {\mathrm{V}}^{\prime } \) be a neighbourhood of \( f\left( x\right) \) in \( {\mathrm{X}}^{\prime } \). Since \( f \) is continuous at \( x,{\overrightarrow{f}}^{\prime }\left( {\mathrm{V}}^{\prime }\right) \) is a neighbourhood of \( x \) in \( \mathrm{X} \). Hence \( {\overrightarrow{f}}^{\prime }\left( {\math... | Yes |
Proposition 2. Let \( \\mathrm{X},{\\mathrm{X}}^{\\prime },{\\mathrm{X}}^{\\prime \\prime } \) be three topological spaces; let \( f \) be a mapping of \( \\mathbf{X} \) into \( {\\mathbf{X}}^{\\prime } \), continuous at \( x \\in \\mathbf{X} \) ; let \( g \) be a mapping of \( {\\mathbf{X}}^{\\prime } \) into \( {\\ma... | Let \( {\\mathrm{V}}^{\\prime \\prime } \) be a neighbourhood of \( h\\left( x\\right) = g\\left( {f\\left( x\\right) }\\right) \) in \( {\\mathrm{X}}^{\\prime \\prime } \) . Since \( g \) is continuous at \( f\\left( x\\right) \) it follows that \( {g}^{\\prime }\\left( {\\mathrm{V}}^{\\prime \\prime }\\right) \) is a... | Yes |
Proposition 3. Given two topologies \( {\% }_{1},{\% }_{2} \) on a set \( \mathrm{X} \), the following statements are equivalent :\n\na) \( {\mathfrak{T}}_{1} \) is finer than \( {\mathfrak{T}}_{2} \) .\n\nb) For each \( x \in \mathrm{X} \), each neighbourhood of \( x \) for \( {\mathcal{C}}_{2} \) is a neighbourhood o... | Example. * In Hilbert space \( \mathrm{H} \) consisting of sequences \( x = \left( {x}_{n}\right) \) of real numbers such that\n\n\[ \parallel x{\parallel }^{2} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{x}_{n}^{2} < + \infty \]\n\nthe neighbourhoods of a point \( {\mathbf{x}}_{\mathbf{0}} \) in the strong topology on... | No |
Proposition 4. Let \( \mathrm{X} \) be a set, let \( {\left( {\mathrm{Y}}_{\mathrm{t}}\right) }_{\mathrm{t} \in \mathrm{I}} \) be a family of topological spaces, and for each \( \mathfrak{l} \in \mathbf{I} \) let \( {f}_{\mathfrak{i}} \) be a mapping of \( \mathbf{X} \) into \( {\mathbf{Y}}_{\mathfrak{l}} \) . Let \( \... | Let \( \mathfrak{O} \) be the set of all unions of sets belonging to \( \mathfrak{B} \) ; clearly \( \mathfrak{O} \) satisfies axiom \( \left( {\mathrm{O}}_{\mathrm{I}}\right) \) since formation of unions is associative; and \( \mathfrak{O} \) satisfies axiom \( \left( {\mathrm{O}}_{\mathrm{{II}}}\right) \) by reason o... | Yes |
Proposition 6. Let \( \mathbf{X} \) be a set, let \( {\left( {\mathbf{Y}}_{\mathfrak{t}}\right) }_{\mathfrak{t} \in \mathbf{I}} \) be a family of topological spaces, and for each \( \mathfrak{x} \in \mathbf{I} \) let \( {f}_{\mathfrak{i}} \) be a mapping of \( {\mathrm{Y}}_{\mathfrak{i}} \) into \( \mathrm{X} \) . Let ... | It is immediately verified that \( \mathfrak{O} \) satisfies the axioms \( \left( {\mathrm{O}}_{\mathrm{I}}\right) \) and \( \left( {\mathrm{O}}_{\mathrm{{II}}}\right) \) . [Set Theory, R, § 4, formulae (34) and (46)]. We shall prove the last assertion of the proposition, which implies the other assertions by reason of... | Yes |
Proposition 8. Let \( \mathrm{X} \) be a set, \( {\left( {\mathrm{X}}_{\lambda }\right) }_{\lambda \in \mathrm{L}} \) a family of subsets of \( \mathrm{X} \) . Suppose each \( {\mathrm{X}}_{\lambda } \) carries a topology \( {\mathcal{C}}_{\lambda } \) such that, for each pair of indices \( \left( {\lambda ,\mu }\right... | In view of Proposition 6 and its corollary, it is enough to show that for each \( \lambda \) and each subset \( {\mathrm{A}}_{\lambda } \) of \( {\mathrm{X}}_{\lambda } \) the following statements are equivalent: (i) \( {\mathrm{A}}_{\lambda } \) is open (resp. closed) in the topology \( {\mathfrak{G}}_{\lambda } \). (... | Yes |
Proposition 1. If \( \mathrm{A} \) and \( \mathrm{B} \) are two subsets of a topological space \( \mathrm{X} \), and \( \mathrm{B} \subset \mathrm{A} \), then the closure of \( \mathrm{B} \) in the subspace \( \mathrm{A} \) is the intersection with \( \mathrm{A} \) of the closure \( \mathbf{B} \) of \( \mathbf{B} \) in... | If \( x \in \mathrm{A} \), every neighbourhood of \( x \) in \( \mathrm{A} \) is of the form \( \mathrm{V} \cap \mathrm{A} \), where \( \mathbf{V} \) is a neighbourhood of \( x \) in \( \mathbf{X} \) . Since \( \mathbf{V} \cap \mathbf{B} = \left( {\mathbf{V} \cap \mathbf{A}}\right) \cap \mathbf{B} \), it follows that \... | Yes |
Proposition 2. Let \( \mathrm{A} \) be a dense subset of a topological space \( \mathrm{X} \) ; then for each \( x \in \mathrm{A} \) and each neighbourhood \( \mathrm{V} \) of \( x \) relative to \( \mathrm{A} \), the closure \( \overline{\mathrm{V}} \) of \( \mathrm{V} \) in \( \mathrm{X} \) is a neighbourhood of \( x... | For \( \mathrm{V} \) contains \( \mathrm{U} \cap \mathrm{A} \), where \( \mathrm{U} \) is an open subset of \( \mathrm{X} \) which contains \( x \), hence \( \overline{\mathrm{V}} \) contains \( \mathrm{U} \cap \overline{\mathrm{A}} = \mathrm{U} \) (§ 1, no. 6, Proposition 5). | Yes |
Proposition 3. Let \( {\left( {\mathbf{A}}_{\imath }\right) }_{\imath \in \mathbf{I}} \) be a family of subsets of a topological space \( \mathbf{X} \) , such that one of the following properties holds: a) The interiors of the \( {\mathrm{A}}_{\mathrm{t}} \) cover \( \mathrm{X} \) . b) \( {\left( {\mathbf{A}}_{\mathbf{... | Clearly if \( \mathrm{B} \) is open (resp. closed) in \( \mathrm{X} \), then \( \mathrm{B} \cap {\mathrm{A}}_{\mathrm{t}} \) is open (resp. closed) in A.. Conversely, suppose first that condition a) is satisfied; since \( \left( {\complement B}\right) \cap {A}_{t} = {A}_{t} - \left( {B \cap {A}_{t}}\right) \), it is en... | Yes |
Proposition 4. Let \( {\left( {\mathbf{A}}_{t}\right) }_{t \in \mathbf{I}} \) be a family of subsets of a topological space \( \mathbf{X} \) whose interiors cover \( \mathrm{X} \), or which is a locally finite closed covering of \( \mathrm{X} \) . Let \( f \) be a mapping of \( \mathbf{X} \) into a topological space \(... | For if \( {\mathrm{F}}^{\prime } \) is a closed subset of \( {\mathrm{X}}^{\prime } \) and if \( \mathrm{F} = {\overrightarrow{f}}^{\prime }\left( {\mathrm{F}}^{\prime }\right) \), then \( \mathrm{F} \cap {\mathrm{A}}^{\prime } \) is closed in \( {A}_{t} \) for each \( \iota \in I\left( {§2,\text{no.}I\text{, Theorem}I... | No |
Proposition 5. For a subset \( \mathbf{L} \) of a topological space \( \mathbf{X} \), the following properties are equivalent:\n\na) \( \mathrm{L} \) is locally closed in \( \mathrm{X} \) .\n\nb) L is the intersection of an open subset and a closed subset of \( \mathbf{X} \) .\n\nc) \( \mathrm{L} \) is open in its clos... | If \( L \) is locally closed, then, for each \( x \in L \), there is an open neighbourhood \( {\mathrm{V}}_{x} \) of \( x \) in \( \mathrm{X} \) such that \( \mathrm{L} \cap {\mathrm{V}}_{x} \) is closed in \( {\mathrm{V}}_{x};\mathrm{U} = \mathop{\bigcup }\limits_{{x \in \mathrm{L}}}{\mathrm{V}}_{x} \) is open in \( \... | Yes |
Proposition 6. Let \( \mathrm{X} \) be a topological space, \( \mathrm{R} \) an equivalence relation on \( \mathrm{X},\varphi \) the canonical mapping of \( \mathrm{X} \) onto \( \mathrm{X}/\mathrm{R} \) ; then a mapping \( f \) of \( \mathrm{X}/\mathrm{R} \) into a topological space \( \mathbf{Y} \) is continuous if a... | This is a particular case of \( §2 \), no. 4, Proposition 6 ; it expresses the fact that the quotient topology is the final topology for the mapping \( \varphi \) . | No |
Proposition 8. Let \( f = \psi \circ g \circ \varphi \) be the canonical decomposition of a continuous mapping \( f : \mathrm{X} \rightarrow \mathrm{Y} \), and let \( \mathrm{R} \) denote the equivalence relation\n\n\[ f\left( x\right) = f\left( y\right) \]\n\nThen the following three conditions are equivalent:\n\na) \... | For the condition b) [resp. c)] expresses that the image under \( g \) of every open (resp. closed) set in \( \mathrm{X}/\mathrm{R} \) is an open (resp. closed) set in \( f\left( \mathrm{X}\right) \) . | Yes |
Proposition 9. Let \( f : \mathrm{X} \rightarrow \mathrm{Y} \) be a continuous surjection, and let \( \mathrm{R} \) denote the equivalence relation \( f\left( x\right) = f\left( y\right) \) . If there is a continuous section \( s : \mathrm{Y} \rightarrow \mathrm{X} \) associated with \( f \) (Set Theory, Chapter II,§ 3... | For if \( \varphi : \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) is the canonical mapping, then \( g \) and \( \varphi \circ s \) are bijective, continuous and inverse to each other; likewise \( s \) and the restriction of \( f \) to \( s\left( \mathbf{Y}\right) \) are bijective, continuous and inverse to each other... | Yes |
Proposition 10. The canonical bijection \( h : \mathrm{A}/{\mathrm{R}}_{\mathrm{A}} \rightarrow f\left( \mathrm{\;A}\right) \) is continuous. Furthermore, the following three statements are equivalent :\n\na) \( h \) is a homeomorphism.\n\nb) Every open subset of \( \mathrm{A} \) which is saturated with respect to \( {... | The first part of the proposition is immediate (no. 5). The second part follows from Proposition 8 of no. 5 : if \( \mathbf{B} \) is an open (resp. closed) subset of \( \mathrm{A} \) which is saturated with respect to \( {\mathrm{R}}_{\mathrm{A}} \), and \( g\left( \mathrm{\;B}\right) = f\left( \mathrm{\;B}\right) \) i... | Yes |
Corollary 1. If \( \mathrm{A} \) is an open (resp. closed) subset of \( \mathrm{X} \) which is saturated with respect to \( \mathrm{R} \), then the canonical mapping \( h : \mathrm{A}/{\mathrm{R}}_{\mathrm{A}} \rightarrow f\left( \mathrm{\;A}\right) \) is a homeomorphism. | For if \( \mathrm{A} \) is open (resp. closed) in \( \mathrm{X} \) and saturated with respect to \( \mathrm{R} \), and if \( \mathrm{B} \subset \mathrm{A} \) is open (resp. closed) in \( \mathrm{A} \) and saturated with respect to \( {R}_{A} \), then \( B \) is open (resp. closed) in \( X \) and saturated with respect ... | No |
Corollary 2. If there is a continuous mapping \( u : \mathrm{X} \rightarrow \mathrm{A} \) such that \( u\left( x\right) \) is congruent to \( x{\;\operatorname{mod}\;\mathrm{R}} \) for each \( x \in \mathrm{X} \), then \( f\left( \mathrm{\;A}\right) = \mathrm{X}/\mathrm{R} \) and the canonical mapping \( h : \mathrm{A}... | Since each equivalence class mod \( R \) meets \( A \), the canonical image of \( \mathrm{A}/{\mathrm{R}}_{\mathrm{A}} \) in \( \mathrm{X}/\mathrm{R} \) is the whole of \( \mathrm{X}/\mathrm{R} \) ; on the other hand, if \( \mathrm{U} \) is open in \( \mathrm{A} \) and is saturated with respect to \( {\mathrm{R}}_{\mat... | No |
Proposition 1. Let \( f = \left( {f}_{i}\right) \) be a mapping of a topological space \( \mathrm{Y} \) into a product space \( \mathrm{X} = \mathop{\prod }\limits_{{i \in \mathrm{I}}}{\mathrm{X}}_{i} \). Then \( f \) is continuous at a point \( a \in \mathrm{Y} \) if and only if \( {f}_{\iota } \) is continuous at a f... | Since \( {f}_{i} = {\operatorname{pr}}_{i} \circ f \), this is just a particular case of Proposition 4 of \( §2 \) , no. 3. | Yes |
Corollary 2. Let \( \mathrm{X},\mathrm{Y} \) be two topological spaces. In order that a mapping \( f : \mathrm{X} \rightarrow \mathrm{Y} \) should be continuous it is necessary and sufficient that the mapping \( g : x \rightarrow \left( {x, f\left( x\right) }\right) \) is a homeomorphism of \( \mathrm{X} \) onto the gr... | Since \( f = {\mathrm{{pr}}}_{2} \circ g \), the condition is sufficient. It is also necessary, for if \( f \) is continuous, then \( g \) is bijective and continuous (Proposition 1) and the inverse of \( g \) is the restriction of \( {\mathrm{{pr}}}_{1} \) to \( \mathrm{G} \), which is continuous (cf. Set Theory, Chap... | Yes |
Proposition 2 (Associativity of topological products). Let \( {\left( {\mathbf{X}}_{\imath }\right) }_{\imath \in \mathbf{I}} \) be a family of topological spaces, \( {\left( {\mathrm{J}}_{\mathrm{x}}\right) }_{\mathrm{x} \in \mathrm{K}} \) a partition of the set \( \mathrm{I} \), and for each \( x \in \mathrm{K} \) le... | This is a particular case of transitivity of initial topologies \( (§2 \), no. 3, Proposition 5; cf. Set Theory, Chapter IV, \( §2 \), no. 4, criterion CST 13). | No |
Proposition 4. Let \( {\mathrm{X}}_{1},{\mathrm{X}}_{2} \) be two topological spaces; then for each \( {a}_{1} \in {\mathrm{X}}_{1} \) , the mapping \( {x}_{2} \rightarrow \left( {{a}_{1},{x}_{2}}\right) \) is a homeomorphism of \( {\mathrm{X}}_{2} \) onto the subspace \( \left\{ {a}_{1}\right\} \times {\mathrm{X}}_{2}... | This is a particular case of Corollary 1 of Proposition 1 of no. I applied to the constant function \( {x}_{2} \rightarrow {a}_{1} \) . | No |
Proposition 5. The projection of an open set \( \mathrm{U} \) of the product \( {\mathrm{X}}_{1} \times {\mathrm{X}}_{2} \) onto either factor is an open set. | For example, we have \( {\operatorname{pr}}_{2}\mathrm{U} = \mathop{\bigcup }\limits_{{{x}_{1} \in {\mathrm{X}}_{1}}}\mathrm{U}\left( {x}_{1}\right) \), and the proposition follows from the Corollary to Proposition 4 and axiom \( \left( {\mathrm{O}}_{\mathrm{I}}\right) \) . | No |
Proposition 6. Let \( {\mathrm{X}}_{1},{\mathrm{X}}_{2},\mathrm{Y} \) be three topological spaces, \( f \) a mapping of the product space \( {\mathrm{X}}_{1} \times {\mathrm{X}}_{2} \) into \( \mathrm{Y} \) . If \( f \) is continuous at the point \( \left( {{a}_{1},{a}_{2}}\right) \) then the partial mapping \( {x}_{2}... | For this mapping is the composition of \( f \) and the mapping \( {x}_{2} \rightarrow \left( {{a}_{1},{x}_{2}}\right) \) ; hence the result follows from Proposition 4. | Yes |
Proposition 7. In a product space \( \coprod {\mathbf{X}}_{t} \), the closure of a product of sets \( \mathop{\prod }\limits_{{i \in \mathbf{I}}}{\mathrm{A}}_{i} \) is the same as the product \( \mathop{\prod }\limits_{{i \in \mathbf{I}}}{\overline{\mathrm{A}}}_{i}^{i \in \mathbf{I}} \) of their closures. | Suppose that \( a = \left( {a}_{t}\right) \) lies in the closure of \( \mathop{\prod }\limits_{t}{\mathrm{A}}_{t} \) ; then for each \( x \in \mathrm{I} \) , \( {a}_{x} = {\operatorname{pr}}_{x}a \) is in the closure of \( {A}_{x} \) because of the continuity of \( {\operatorname{pr}}_{x}(§2 \) , no. 1, Theorem 1) and ... | Yes |
Proposition 8. Let \( a = \left( {a}_{\iota }\right) \) be any point of a product space \( \mathrm{X} = \mathop{\prod }\limits_{{\iota \in \mathrm{I}}}{\mathrm{X}}_{\iota } \) ; then the set \( \mathrm{D} \) of points \( x \in \mathrm{X} \) such that \( {\operatorname{pr}}_{\iota }x = {a}_{\iota } \) except for a finit... | For each \( x \in \mathrm{X} \) and each elementary set \( \mathrm{V} = \mathop{\prod }\limits_{{i \in \mathrm{I}}}{\mathrm{U}}_{i} \) which contains \( x \) , we have \( {U}_{t} = {X}_{t} \) except for indices \( \iota \) belonging to a finite subset \( J \) of \( \mathrm{I} \) ; if we take \( {y}_{\mathrm{t}} = {x}_{... | Yes |
Proposition 9. Let \( \mathrm{I} \) be a directed set and \( \mathrm{J} \) a cofinal subset of \( \mathrm{I} \) . Let \( \left( {{\mathrm{X}}_{\alpha },{f}_{\alpha \beta }}\right) \) be an inverse system of topological spaces indexed by \( \mathrm{I} \) ; let \( \mathrm{X} = \mathop{\lim }\limits_{ \leftarrow }{\mathrm... | From \( §2, \) no. 3 we know that the finite intersections of sets of the form \( {\overrightarrow{f}}_{\alpha }\left( {\mathrm{U}}_{\alpha }\right) \) \( \left( {\alpha \in \mathrm{I},{\mathrm{U}}_{\alpha }\text{open in}{\mathrm{X}}_{\alpha }}\right) \) form a base of the topology of \( \mathrm{X} \) . If \( {\left( {... | Yes |
Proposition 1. Let \( \mathrm{X},{\mathrm{X}}^{\prime },{\mathrm{X}}^{\prime \prime } \) be three topological spaces, and let \( f \) : \( \mathrm{X} \rightarrow {\mathrm{X}}^{\prime }, g : {\mathrm{X}}^{\prime } \rightarrow {\mathrm{X}}^{\prime \prime } \) be two mappings. Then:\n\na) If \( f \) and \( g \) are open (... | From Definition 1. a) follows immediately. To prove b) it is enough to remark that every open (resp. closed) subset \( {\mathrm{A}}^{\prime } \) of \( {\mathrm{X}}^{\prime } \) can be written as \( f\left( \mathrm{\;A}\right) \), where \( \mathrm{A} = {f}^{\prime }\left( {\mathrm{A}}^{\prime }\right) \) is open (resp. ... | Yes |
a) If \( f \) is open (resp. closed), \( {f}_{\mathrm{T}} \) is open (resp. closed). | a) If \( \mathrm{A} \) is an open (resp. closed) subset of \( {\overrightarrow{f}}^{\prime }\left( \mathbf{T}\right) \), then there is an open (resp. closed) subset \( \mathbf{B} \) of \( \mathbf{X} \) such that \( \mathbf{A} = \mathbf{B} \cap {\bar{f}}^{1}\left( \mathbf{T}\right) \), and therefore \( {f}_{\mathrm{T}}\... | Yes |
Proposition 4. Let \( R \) be an open (resp. closed) equivalence relation on a topological space \( \mathrm{X} \), and \( f \) the canonical mapping \( \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \). Let \( \mathrm{A} \) be a subset of \( \mathrm{X} \) and suppose that one of the following two conditions is satisfied:... | Consider the commutative diagram (1) of \( §3 \), no. 6, which gives the canonical decomposition of \( f \circ j \). Under condition a), \( j \) is open (resp. closed) and so is \( f \) by hypothesis; hence \( f \circ j \) is open (resp. closed) [no. 1, Proposition 1 a)], and the result follows from Proposition 3. Unde... | Yes |
Proposition 5. Let \( \mathbf{X},\mathbf{Y} \) be two topological spaces, \( f \) a mapping of \( \mathbf{X} \) into \( \mathrm{Y},\mathfrak{B} \) a base of the topology of \( \dot{\mathrm{X}} \) . Then the following statements are equivalent :\n\na) \( f \) is an open mapping.\n\nb) For each \( \mathbf{U} \in \mathfra... | The equivalence of a) and b) follows immediately from the definitions and from \( \left( {\mathrm{O}}_{\mathrm{I}}\right) \) ; the equivalence of a) and c) is a consequence of Proposition I of \( § \) I, no. 2. | No |
Proposition 6. Let \( \mathbf{R} \) be an equivalence relation on a topological space \( \mathbf{X} \) ; then the following three conditions are equivalent:\n\na) The relation \( \mathbf{R} \) is open.\n\nb) The interior of each subset which is saturated with respect to \( \mathbf{R} \) is saturated with respect to \( ... | By taking complements ( \( § \) 1, no. 6, formula (2)) we see that b) and c) are equivalent. Let us show that b) implies a): suppose condition b) is satisfied and let \( U \) be an open subset of \( X, V \) its saturation with respect to \( \mathrm{R} \) ; then \( \dot{\mathrm{V}} \supset \mathrm{U} \), and since by hy... | Yes |
Proposition 7. Let \( \mathrm{R} \) be an open equivalence relation on a topological space \( \mathrm{X} \) , and let \( \varphi : \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) be the canonical mapping. If \( \mathrm{A} \) is any subset of \( \mathrm{X} \) which is saturated with respect to \( \mathbf{R} \), then the... | Each of the two assertions of the proposition can be deduced from the other by taking complements and using formula (2) of \( § \) 1, no. 6 and the fact that if \( B \) is a saturated subset of \( X \) then \( \varphi \left( {CB}\right) = {C\varphi }\left( B\right) \) . By virtue of Proposition 6, \( \overline{\mathrm{... | Yes |
Proposition 8. Let \( {\left( {\mathrm{X}}_{\mathrm{t}}\right) }_{\mathrm{t} \in \mathrm{I}},{\left( {\mathrm{Y}}_{\mathrm{t}}\right) }_{\mathrm{t} \in \mathrm{I}} \) be two families of topological spaces indexed by the same set \( \mathrm{I} \) . For each \( \mathrm{t} \in \mathrm{I} \) let \( {f}_{\mathrm{t}} \) be a... | By virtue of Proposition 5 we need only prove that the image under \( f \) of any elementary set \( \mathop{\prod }\limits_{{i \in I}}{\mathrm{\;A}}_{i} \) in \( \mathop{\prod }\limits_{{i \in I}}{\mathrm{X}}_{i} \) is open in \( \mathop{\prod }\limits_{{i \in I}}{\mathrm{Y}}_{i} \) . But this image is \( \mathop{\prod... | Yes |
Proposition 9. Let \( \mathrm{X},{\mathrm{X}}^{\prime } \) be two topological spaces. A necessary and sufficient condition for a mapping \( f : \mathrm{X} \rightarrow {\mathrm{X}}^{\prime } \) to be continuous and closed is that \( f\left( \overline{\mathrm{A}}\right) = \overline{f\left( \mathrm{\;A}\right) } \) for ev... | The condition is sufficient, for it obviously implies that \( f \) is closed, and it also implies that \( f \) is continuous by reason of \( §2 \), no. 1, Theorem 1. Conversely, if \( f \) is continuous and closed, we have \( f\left( \mathrm{\;A}\right) \subset f\left( \overline{\mathrm{A}}\right) \subset \overline{f\l... | Yes |
Proposition 10. Let \( \mathrm{R} \) be an equivalence relation on a topological space \( \mathrm{X} \) . Then \( \mathrm{R} \) is closed if and only if every equivalence class \( \mathrm{M} \) mod \( \mathrm{R} \) possesses a fundamental system of neighbourhoods which are saturated with respect to \( \mathrm{R} \) . | Suppose \( R \) is closed, and let \( U \) be an arbitrary open neighbourhood of \( \mathbf{M} \) ; since \( \mathbf{F} = \mathbf{C}\mathbf{U} \) is closed in \( \mathbf{X} \), the saturation \( \mathbf{S} \) of \( \mathbf{F} \) with respect to \( \mathrm{R} \) is closed in \( \mathrm{X} \) . Since \( \mathrm{M} \) is ... | Yes |
Corollary 2. A set \( \Phi \) of filters on a non-empty set \( \mathrm{X} \) has a least upper bound in the set of all filters on \( \mathbf{X} \) if and only if, for all finite sequences \( {\left( {\mathfrak{F}}_{i}\right) }_{1 \leq i \leq n} \) of elements of \( \Phi \) and all \( {\mathrm{A}}_{i} \in {\mathfrak{F}}... | For this condition expresses that the union \( \Theta \) of the filters \( \mathfrak{F} \Subset \Phi \) satisfies the condition of Proposition 1. | No |
Corollary 3. The ordered set of all filters on a non-empty set \( \mathbf{X} \) is inductive. | For every linearly ordered set \( \Phi \) of filters on \( \mathrm{X} \) satisfies the condition of Corollary 2 of Proposition 1, since the sets \( {A}_{t} \) all belong to the same \( {\mathfrak{F}}_{i} \) by hypothesis, and we can apply \( \left( {\mathbf{F}}_{\mathbf{{II}}}\right) \) . | No |
Proposition 3. A subset \( \mathfrak{B} \) of a filter \( \mathfrak{F} \) on \( \mathrm{X} \) is a base of \( \mathfrak{F} \) if and only if every set of \( \mathfrak{F} \) contains a set of \( \mathfrak{B} \) . | If \( \mathfrak{B} \) is a base of \( \mathfrak{F} \), then clearly every set of \( \mathfrak{F} \) contains a set of \( \mathfrak{B} \) : conversely, if every set of \( \mathfrak{F} \) contains a set of \( \mathfrak{B} \), then the set of subsets of \( \mathrm{X} \) containing a set of \( \mathfrak{B} \) coincides wit... | Yes |
Proposition 4. On a set \( \mathrm{X} \), a filter \( {\mathfrak{F}}^{\prime } \) with base \( {\mathfrak{B}}^{\prime } \) is finer than a filter \( \mathfrak{F} \) with base \( \mathfrak{B} \) if and only if every set of \( \mathfrak{B} \) contains a set of \( {\mathfrak{B}}^{\prime } \) . | This is an immediate consequence of Definitions 2 and 3. | No |
Proposition 5. Let \( \mathfrak{F} \) be an ultrafilter on a set \( \mathrm{X} \). If \( \mathrm{A} \) and \( \mathrm{B} \) are two subsets of \( \mathrm{X} \) such that \( \mathrm{A} \cup \mathrm{B} \in \mathfrak{F} \), then either \( \mathrm{A} \in \mathfrak{F} \) or \( \mathrm{B} \in \mathfrak{F} \). If the proposit... | Let \( \Theta \) be the set of subsets \( M \) of \( X \) such that \( \mathrm{A} \cup \mathrm{M} \in \mathfrak{F} \). It is straightforward to check that \( \circledast \) is a filter on \( \mathrm{X} \), and \( \Theta \) is strictly finer than \( \mathfrak{F} \), since \( \mathrm{B} \in \Theta \) ; but this contradic... | No |
Proposition 6. Let \( \circledast \) be a subbase of a filter on a set \( \mathrm{X} \) . If for each subset \( \mathrm{Y} \) of \( \mathrm{X} \) we have either \( \mathrm{Y} \in \mathcal{G} \) or \( \complement \mathrm{Y} \in \mathcal{G} \), then \( \mathcal{G} \) is an ultrafilter on \( \mathrm{X} \). | Let \( \mathfrak{F} \) be a filter containing \( \Theta \) (there is one, by hypothesis); then \( \mathfrak{F} \) coincides with \( \mathcal{O} \) ; for if \( \mathrm{Y} \in \mathfrak{F} \) then \( \llbracket \mathrm{Y} \in \mathfrak{F} \) ; hence \( \llbracket \mathrm{Y} \in \mathcal{O} \) and therefore \( \mathrm{Y} ... | No |
Proposition 7. Every filter \( \mathfrak{F} \) on a set \( \mathrm{X} \) is the intersection of the ultrafilters finer than \( \delta \) . | Clearly this intersection contains \( \mathfrak{F} \) . Conversely, let A be a subset of \( \mathrm{X} \) which does not belong to \( \mathfrak{F} \), and let \( {\mathrm{A}}^{\prime } \) denote \( ▱\mathrm{A} \) ; A contains no set of \( \mathfrak{F} \) ; hence every \( M \in \mathfrak{F} \) meets \( {A}^{\prime } \) ... | Yes |
Proposition 8. Let \( \mathfrak{F} \) be a filter on a set \( \mathrm{X} \) and \( \mathrm{A} \) a subset of \( \mathrm{X} \) . Then the trace \( {\mathfrak{F}}_{\mathbf{A}} \) of \( \mathfrak{F} \) on \( \mathbf{A} \) is a filter if and only if each set of \( \mathfrak{F} \) meets \( \mathbf{A} \) . | Since \( \left( {\mathbf{M} \cap \mathbf{N}}\right) \cap \mathbf{A} = \left( {\mathbf{M} \cap \mathbf{A}}\right) \cap \left( {\mathbf{N} \cap \mathbf{A}}\right) \) we see that \( {\mathfrak{F}}_{\mathbf{A}} \) satisfies \( \left( {\mathbf{F}}_{\mathbf{{II}}}\right) \) ; again, if \( \mathrm{M} \cap \mathrm{A} \subset \... | Yes |
Proposition 9. An ultrafilter \( \mathfrak{u} \) on a set \( \mathrm{X} \) induces a filter on a subset \( \mathrm{A} \) of \( \mathrm{X} \) if and only if \( \mathrm{A} \in \mathfrak{u} \) ; and if this condition is satisfied then \( {\mathfrak{u}}_{\mathrm{A}} \) is an ultrafilter on \( \mathrm{A} \) . | This is an immediate consequence of Propositions 5 and 6 of no. 4. | No |
Proposition 10. If \( \mathfrak{B} \) is an ultrafilter base on a set \( \mathrm{X} \) and if \( f \) is a mapping of \( \mathbf{X} \) into a set \( {\mathbf{X}}^{\prime } \), than \( f\left( \mathfrak{B}\right) \) is an ultrafilter base on \( {\mathbf{X}}^{\prime } \) . | Let \( {\mathbf{M}}^{\prime } \) be a subset of \( {\mathbf{X}}^{\prime } \) . If \( {\bar{f}}^{\prime }\left( {\mathbf{M}}^{\prime }\right) \) contains a set \( \mathbf{M} \) of \( \mathfrak{B} \), then \( {\mathrm{M}}^{\prime } \) contains \( f\left( \mathrm{M}\right) \) ; if not, then \( {\int }^{-1}\left( {\mathrm{... | No |
Proposition 1. A filter base \( \mathfrak{B} \) on a topological space \( \mathrm{X} \) converges to \( x \) if and only if every set of a fundamental system of neighbourhoods of \( x \) contains a set of \( \mathfrak{B} \) . | In accordance with the terminology introduced in \( § \) 1, no. 2 we can state Proposition 1 in the following way : \( \mathfrak{B} \) converges to \( x \) if and only if there are sets of \( \mathfrak{B} \) as near as we please to \( x \) . | No |
Proposition 2. A filter \( \mathfrak{F} \) on a topological space \( \mathbf{X} \) converges to a point \( x \) if and only if every ultrafilter which is finer than \( \mathfrak{F} \) converges to \( x \) . | This is an immediate consequence of the preceding remarks and Proposition \( 7 \) of \( §6, \) no. 4. | No |
Proposition 3. A point \( x \) is a cluster point of a filter base \( \mathfrak{B} \) if and only if every set of a fundamental system of neighbourhoods of \( x \) meets every set of \( \mathfrak{B} \) . | This follows immediately from the definitions. | No |
Proposition 6. Let \( \mathfrak{B} \) be a filter base on a subset \( \mathbf{A} \) of a topological space \( \mathbf{X} \) . Then every cluster point of \( \mathfrak{B} \) in \( \mathrm{X} \) belongs to \( \overline{\mathrm{A}} \) ; and conversely every point of \( \overline{\mathrm{A}} \) is a limit point of a filter... | The first assertion is trivial; on the other hand, if \( x \in \overline{\mathrm{A}} \), the trace on \( \mathrm{A} \) of the neighbourhood filter of \( x \) in \( \mathrm{X} \) is a filter on \( \mathrm{A} \) which evidently converges to \( x \) . | No |
Proposition 8. Let \( f \) be a mapping of a set \( \mathbf{X} \) into a topological space \( \mathbf{Y} \) ; then \( y \in \mathrm{Y} \) is a cluster point of \( f \) with respect to \( \% \) if and only if there is a filter ( \( y \) on \( \mathrm{X} \) which is finer than \( \mathfrak{F} \) and such that \( y \) is ... | For if \( y \) is a cluster point of \( f \) with respect to \( \mathfrak{F} \), and if \( \mathfrak{B} \) is the neighbourhood filter of \( y \), then \( {\bar{f}}^{2}\left( \mathfrak{V}\right) \) is a filter base on \( \mathbf{X} \) since every set of \( \overrightarrow{f}\left( \mathfrak{V}\right) \) meets every set... | Yes |
Let \( \mathrm{X},\mathrm{Y} \) be two topological spaces, \( f \) a mapping of \( \mathrm{X} \) into \( \mathrm{Y} \) which is continuous at a point \( a \in \mathrm{X} \); then, for every filter base \( \mathfrak{B} \) on \( \mathrm{X} \) which converges to a, the filter base \( f\left( \mathfrak{B}\right) \) converg... | The first assertion is an immediate consequence of Proposition 9. To prove the second, suppose that \( f \) is not continuous at \( a \); then there is a neighbourhood \( \mathrm{W} \) of \( f\left( a\right) \) in \( \mathrm{Y} \) such that \( {\bar{f}}^{1}\left( \mathrm{\;W}\right) \) does not belong to the filter \( ... | Yes |
Proposition 10. Let \( \mathrm{X} \) be a set, let \( {\left( {\mathrm{Y}}_{t}\right) }_{t \in \mathrm{I}} \) be a family of topological spaces, and for each \( \iota \in \mathbf{I} \) let \( {f}_{\iota } \) be a mapping of \( \ddot{\mathbf{X}} \) into \( {\check{\mathbf{Y}}}_{\iota } \) . Let \( \mathbf{X} \) be given... | The condition is necessary since the \( {f}_{t} \) are continuous (no. 4, Proposition 9, Corollary 1). Conversely, suppose that the condition is satisfied, and let \( \mathrm{V} \) be an open neighbourhood of \( a \) in \( \mathrm{X} \) . By the definition of To (§ 2, no. 3, Proposition 4) there is a finite subset \( J... | Yes |
Proposition 11. Let \( \mathrm{R} \) be an open equivalence relation on a topological space \( \mathrm{X} \) and let \( \varphi \) be the canonical mapping \( \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) . Then for each \( x \in \mathrm{X} \) and each filter base \( {\mathfrak{B}}^{\prime } \) on \( \mathrm{X}/\math... | If \( \mathbf{U} \) is any neighbourhood of \( x \) in \( \mathbf{X} \), then \( \varphi \left( \mathbf{U}\right) \) is a neighbourhood of \( \varphi \left( x\right) \) in \( \mathrm{X}/\mathrm{R} \) (§ 5, no. 3, Proposition 5), hence there is a set \( {\mathbf{M}}^{\prime } \in {\mathfrak{B}}^{\prime } \) such that \(... | Yes |
Proposition 2. Let \( f, g \) be two continuous mappings of a topological space \( \mathbf{X} \) into a Hausdorff space \( \mathrm{Y} \) ; then the set of all \( x \in \mathrm{X} \) such that \( f\left( x\right) = g\left( x\right) \) is closed in \( \mathrm{X} \). | For this set is the inverse image of the diagonal of \( \mathrm{Y} \times \mathrm{Y} \) under the mapping \( x \rightarrow \left( {f\left( x\right), g\left( x\right) }\right) \), which is continuous \( \left( {§4\text{, no. I, Proposition I}}\right) \) . The result therefore follows from \( \left( {\mathrm{H}}^{\mathrm... | Yes |
Proposition 3. Let \( {\left( {x}_{i}\right) }_{\mathbf{I} \leq i \leq n} \) be a finite family of distinct points of a Hausdorff space \( \mathrm{X} \) ; then each \( {x}_{i} \) has a neighbourhood \( {\mathrm{V}}_{i} \) in \( \mathrm{X} \) such that the \( {\mathrm{V}}_{i}\left( {\mathrm{r} \leq i \leq n}\right) \) a... | The proof is by induction on \( n \) : the case \( n = 2 \) is just the axiom (H). Let then \( {\mathrm{W}}_{i}\left( {\mathrm{I} \leq i \leq n - \mathrm{I}}\right) \) be a neighbourhood of \( {x}_{i} \) such that the \( {\mathrm{W}}_{i} \) are mutually disjoint. On the other hand, for \( \mathrm{I} \leq i \leq n - \ma... | Yes |
Proposition 4. Every finite subset of a Hausdorff space is closed. | For every subset consisting of a single point is closed by reason of axiom \( \left( {\mathrm{H}}^{\mathrm{i}}\right) \) . | No |
Proposition 5. Let \( \mathrm{X} \) be a topological space and suppose that for each pair of distinct points \( x, y \) of \( \mathrm{X} \) there is a continuous mapping \( f \) of \( \mathrm{X} \) into a Hausdorff space \( {\mathrm{X}}^{\prime } \) such that \( f\left( x\right) \neq f\left( y\right) \) . Then \( \math... | Let \( {\mathrm{V}}^{\prime } \) and \( {\mathrm{W}}^{\prime } \) be disjoint neighbourhoods of \( f\left( x\right) \) and \( f\left( y\right) \) respectively in \( {\mathrm{X}}^{\prime } \) ; then \( {\overrightarrow{f}}^{\prime }\left( {\mathrm{V}}^{\prime }\right) \) and \( {\overrightarrow{f}}^{\prime }\left( {\mat... | Yes |
Proposition 6. If every point of a topological space \( \mathbf{X} \) has a closed neighbourhood which is a Hausdorff subspace of \( \mathrm{X} \), then \( \mathrm{X} \) is Hausdorff. | Let \( x \in \mathrm{X} \) and let \( \mathrm{V} \) be a closed neighbourhood of \( x \) in \( \mathrm{X} \) such that the subspace \( \mathrm{V} \) is Hausdorff. Then the closed neighbourhoods of \( x \) in \( \mathrm{V} \) have \( \{ x\} \) as their intersection (axiom \( \left( {\mathrm{H}}^{\prime }\right) \) ; but... | No |
Proposition 7. Every product of Hausdorff spaces is Hausdorff. Conversely, if a product of non-empty spaces is Hausdorff, then each factor is a Hausdorff space. | Let \( \mathrm{X} = \mathop{\prod }\limits_{{i \in \mathrm{I}}}{\mathrm{X}}_{i} \) be a product of topological spaces. Then if \( x, y \) are two distinct points of \( \mathrm{X} \), we have \( {\operatorname{pr}}_{\iota }x \neq {\operatorname{pr}}_{\iota }y \) for some index \( \iota \), and Proposition 5 of no. 1 sho... | Yes |
Corollary 2. Let \( \left( {{\mathrm{X}}_{\alpha },{f}_{\alpha \beta }}\right) \) be an inverse system of topological spaces. If the \( {\mathrm{X}}_{\alpha } \) are Hausdorff, then \( \mathrm{X} = \mathop{\lim }\limits_{ \leftarrow }{\mathrm{X}}_{\alpha } \) is Hausdorff and is a closed subspace of \( \mathop{\prod }\... | The first assertion follows from the fact that \( \mathbf{X} \) is a subspace of the Hausdorff space \( \mathop{\prod }\limits_{\alpha }{\mathrm{X}}_{\alpha } \) (Proposition 7). To show that \( \mathrm{X} \) is closed in the product space, let \( {\mathrm{F}}_{\alpha \beta }\left( {\alpha \leq \beta }\right) \) be the... | Yes |
Proposition 8. A necessary condition for the quotient space \( \mathrm{X}/\mathrm{R} \) to be Hausdorff is that the graph \( \mathrm{C} \) of \( \mathrm{R} \) is closed in \( \mathrm{X} \times \mathrm{X} \) . If the equivalence relation \( \widetilde{\mathrm{R}} \) is open, this condition is also sufficient. | Let \( \varphi : \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) be the canonical mapping; then \( \mathrm{C} \) is the inverse image under \( \varphi \times \varphi : \mathrm{X} \times \mathrm{X} \rightarrow \left( {\mathrm{X}/\mathrm{R}}\right) \times \left( {\mathrm{X}/\mathrm{R}}\right) \) of the diagonal \( \Delta... | Yes |
Proposition 10. If \( \mathrm{X} \) is a Hausdorff space, and if \( \mathrm{X} \) has a continuous section \( s \) with respect to the equivalence relation \( \mathrm{R} \), then \( \mathrm{X}/\mathrm{R} \) is Hausdorff and \( s\left( {\mathrm{X}/\mathrm{R}}\right) \) is closed in \( \mathrm{X} \) . | For \( \left( {§3,\text{ no. }5}\right) \mathrm{X}/\mathrm{R} \) is homeomorphic to the subspace \( s\left( {\mathrm{X}/\mathrm{R}}\right) \) of \( \mathrm{X} \) , which is Hausdorff. Furthermore \( s\left( {\mathrm{X}/\mathrm{R}}\right) \) is the set of all \( x \in \mathrm{X} \) such that \( s\left( {\varphi \left( x... | No |
Proposition 11. The following properties of a topological space \( \mathbf{X} \) are equivalent :\n\n\( \left( {\mathrm{O}}_{\mathrm{{III}}}\right) \) The set of closed neighbourhoods of any point of \( \mathrm{X} \) is a fundamental system of neighbourhoods of the point.\n\n\( \left( {\mathrm{O}}_{\mathrm{{III}}}^{\pr... | \( \left( {\mathrm{O}}_{\mathrm{{III}}}\right) \Rightarrow \left( {\mathrm{O}}_{\mathrm{{III}}}^{\prime }\right) \) : If \( \mathrm{F} \) is closed and \( x \in \mathrm{F} \), then there is a closed neighbourhood \( \mathrm{V} \) of \( x \) contained in the neighbourhood CF of \( x \) ; \( \mathrm{V} \) and CV are neig... | Yes |
Proposition 12. Every subspace of a regular space is regular. | Let \( A \) be a subspace of a regular space \( X \) . Since \( X \) is Hausdorff, so is A (no. 2); on the other hand, every neighbourhood of a point \( x \in \mathrm{A} \) with respect to \( \mathrm{A} \) is of the form \( \mathrm{V} \cap \widetilde{\mathrm{A}} \), where \( \mathrm{V} \) is a neighbourhood of \( x \) ... | No |
Proposition 13. If every point \( x \) of a topological space \( \mathbf{X} \) has a closed neighbourhood which is a regular subspace of \( \mathrm{X} \), then \( \mathrm{X} \) is regular. | \( \mathbf{X} \) is Hausdorff by Proposition 6 of no. 2. Let \( x \) be any point of \( \mathbf{X} \) and let \( \mathrm{V} \) be a closed regular neighbourhood of \( x \) . If \( \mathrm{U} \) is any neighbourhood of \( x \) contained in \( \mathrm{V} \), then \( \mathrm{U} \) is a neighbourhood of \( x \) relative to... | Yes |
Proposition 14. Let \( \mathrm{X} \) be a regular space, \( \mathrm{R} \) a closed equivalence relation on \( \mathrm{X} \) . Then the graph \( \mathrm{C} \) of \( \overline{\mathrm{R}} \) in \( \mathrm{X} \times \mathrm{X} \) is closed. | Let \( \left( {a, b}\right) \) be a point of \( \mathrm{X} \times \mathrm{X} \) in the closure of \( \mathrm{C} \), and let \( \mathrm{V} \) (resp. \( \mathrm{W} \) ) be a closed neighbourhood of \( a \) (resp. a neighbourhood of \( b \) ) in \( \mathrm{X} \) ; then there is a point \( \left( {x, y}\right) \in \mathrm{... | Yes |
Proposition 15. Let \( \mathrm{X} \) be a regular space, \( \mathrm{F} \) a closed subset of \( \mathrm{X},\mathrm{R} \) the equivalence relation on \( \mathrm{X} \) obtained by identifying all the points of \( \mathrm{F} \) [in other words, the equivalence relation whose equivalence classes are \( \mathrm{F} \) (if \(... | Let \( M \) and \( N \) be two distinct equivalence classes in \( X \) . If each of them consists of a single point in the complement of \( \mathrm{F} \), then there exist two disjoint open neighbourhoods of \( M \) and \( N \) in the Hausdorff subspace CF; these are neighbourhoods of \( \mathbf{M} \) and \( \mathbf{N}... | Yes |
Proposition 1. Let \( \mathrm{X} \) be a compact space, \( x \) a point of \( \mathrm{X} \) . In order that a filter base \( \mathfrak{B} \) formed of closed neighbourhoods of \( x \) should be a fundamental system of neighbourhoods of \( x \) it is necessary and sufficient that the intersection of the sets of \( \math... | The condition is necessary since \( X \) is Hausdorff (§ 8, no. 1, Proposition 1). It is sufficient, for it signifies that \( x \) is the only cluster point of \( \mathfrak{B} \) ; hence \( \mathfrak{B} \) converges to \( x \) by the Corollary to Theorem I of no. I. | Yes |
Proposition 2. Let \( \mathrm{X} \) be a compact space and let \( \mathrm{A},\mathrm{B} \) be two disjoint closed subsets of \( \mathrm{X} \) . Then there exist two open sets \( \mathrm{U},\mathrm{V} \), such that \( \mathrm{U} \cap \mathrm{V} = \varnothing \) and \( \mathrm{A} \subset \mathrm{U} \) and \( \mathrm{B} \... | Suppose the conclusion is false. If every neighbourhood \( U \) of \( A \) meets every neighbourhood \( \mathrm{V} \) of \( \mathrm{B} \), then the sets \( \mathrm{U} \cap \mathrm{V} \) form a filter base \( \mathfrak{B} \) on \( \mathrm{X} \), which therefore has a cluster point \( x \in \mathrm{X} \) . Now \( x \) mu... | Yes |
Proposition 3. Every closed subset of a quasi-compact (resp. compact) space is quasi-compact (resp. compact). | This is an immediate consequence of axiom \( \left( {\mathrm{C}}^{\prime \prime }\right) \) if we remark that if \( \mathrm{A} \) is closed in \( \mathrm{X} \) then every set which is closed in \( \mathrm{A} \) is closed in \( \mathrm{X} \) . | No |
Proposition 4. Every compact subset of a Hausdorff space is closed. | Let \( \mathrm{A} \) be a compact subset of a Hausdorff space \( \mathrm{X} \), and let \( x \) be any point of \( \overline{\mathrm{A}} \) ; we have to show that \( x \in \mathrm{A} \) . By hypothesis, every neighbourhood of \( x \) meets \( \mathrm{A} \), and therefore the neighbourhood filter \( \mathfrak{B} \) of \... | Yes |
Proposition 5. The union of a finite family of quasi-compact subsets of a topological space is quasi-compact. | It is sufficient to show that if \( A \) and \( B \) are two quasi-compact subsets of a topological space \( \mathrm{X} \), then \( \mathrm{A} \cup \mathrm{B} \) is quasi-compact. Let \( \mathfrak{R} \) be covering of \( \mathbf{A} \cup \mathbf{B} \) ; then \( \Re \) is a covering of \( \mathbf{A} \) and a covering of ... | Yes |
Proposition 6. If \( \mathrm{X} \) is a Hausdorff space, a subset \( \mathrm{A} \) of \( \mathrm{X} \) is relatively compact if and only if \( \overline{\mathbf{A}} \) is compact. | If \( \mathrm{A} \) is relatively compact, then \( \overline{\mathrm{A}} \) is compact by Proposition 4 and its corollary; the reverse implication is self-evident. | Yes |
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