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Proposition 7. If \( \mathrm{A} \) is a relatively quasi-compact subset of a topological space \( \mathrm{X} \), then every filter base on \( \mathrm{A} \) has a cluster point in \( \mathrm{X} \) . | For if \( \mathrm{A} \subset \mathrm{K} \), where \( \mathrm{K} \) is a quasi-compact subset of \( \mathrm{X} \), then every filter base on \( A \) has a cluster point in \( K \) . | No |
Corollary 2. Every continuous mapping \( f \) of a quasi-compact space \( \mathrm{X} \) into a Hausdorff space \( {\mathbf{X}}^{\prime } \) is closed. If also \( f \) is bijective, then \( f \) is a homeomorphism. | This follows immediately from Corollary \( I \) and Proposition 4 of no. 3. | No |
Corollary 4. Let \( \mathrm{X} \) be a topological space and \( \mathrm{R} \) a Hausdorff equivalence relation on \( \mathrm{X} \). a) If there is a quasi-compact set \( \mathrm{K} \) in \( \mathrm{X} \) which meets every equivalence class mod \( \mathrm{R} \), then \( \mathrm{X}/\mathrm{R} \) is compact and the canoni... | Let \( f \) be the restriction to \( \mathrm{K} \) of the canonical mapping \( \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \). Since \( \mathrm{X}/\mathrm{R} \) is Hausdorff it follows from Corollary 1 that \( \mathrm{X}/\mathrm{R} \) is compact and from Corollary 2 that \( f \) is closed; hence the bijection \( \math... | Yes |
Proposition 8. Let \( \left( {{\mathrm{X}}_{\alpha },{f}_{\alpha \beta }}\right) \) be an inverse system of compact spaces indexed by a directed set I such that \( {f}_{\alpha \alpha } \) is the identity mapping for each \( \alpha \in \mathbf{I} \) . Let \( \dot{\mathrm{X}} = \underline{\lim }{\mathrm{X}}_{\alpha } \) ... | \( \mathrm{X} \) is a closed subspace of \( \mathop{\prod }\limits_{\alpha }{\mathrm{X}}_{\alpha } \) (§ 8, no. 2, Proposition 7, Corollary 2) which is compact by Theorem 3 of no. 5 and Proposition 3 of no. 3. The other assertions are consequences of Set Theory, chap. III, § 7, no. 4, th. I. We apply this theorem by ta... | Yes |
Proposition 9. Every locally compact space is regular. | Let \( \mathrm{X} \) be a locally compact space, then every point \( x \in \mathrm{X} \) has a compact neighbourhood \( \mathrm{V} \) ; since \( \mathrm{X} \) is Hausdorff, \( \mathrm{V} \) is closed (no. 3, Proposition 4). On the other hand, \( \mathrm{V} \) is a regular subspace of \( \mathrm{X} \) (no. 2, Corollary ... | No |
Proposition 10. In a locally compact space \( \mathrm{X} \), every compact set \( \mathrm{K} \) has a fundamental system of compact neighbourhoods. | Let \( U \) be any neighbourhood of \( K \) . For each \( x \in K \) there is a compact neighbourhood \( W\left( x\right) \) of \( x \) contained in U. The interiors of the sets \( \mathrm{W}\left( x\right) \) form an open covering of \( \mathrm{K} \) as \( x \) runs through \( \mathrm{K} \) ; hence there exist a finit... | Yes |
Proposition 11. Let \( \mathrm{X} \) be a locally compact space and \( \mathrm{F} \) a subset of \( \mathrm{X} \) such that \( \mathrm{F} \cap \mathrm{K} \) is compact whenever \( \mathrm{K} \) is a compact subset of \( \mathrm{X} \) . Then \( \mathrm{F} \) is closed in \( \mathbf{X} \) . | In view of Proposition 4 of no. 3, this follows from Proposition 3 a) of \( §3 \), no. 1 . | No |
Proposition 12. In a Hausdorff space \( \mathbf{X} \) , every locally compact subspace \( \mathbf{A} \) is locally closed. | By hypothesis, for every \( x \in \mathrm{A} \) there is a neighbourhood \( \mathrm{V} \) of \( x \) in \( \mathrm{X} \) such that \( \mathrm{V} \cap \mathrm{A} \) is compact and therefore closed in \( \mathrm{V} \) (no. 3, Proposition 4). | Yes |
Proposition 13. Every locally closed subspace of a locally compact space \( \mathbf{X} \) is locally compact. | Let \( \mathrm{A} \) be a locally closed subspace of \( \mathrm{X} \) ; then for each \( x \in \mathrm{A} \) there is a neighbourhood \( \mathrm{U} \) of \( x \) in \( \mathrm{X} \) such that \( \mathrm{U} \cap \mathrm{A} \) is closed in \( \mathrm{U} \) . Let \( \mathrm{V} \subset \mathrm{U} \) be a compact neighbourh... | Yes |
Theorem 1 (no. 1) and Corollary 2 of Theorem 2 (no. 4) do not extend to locally compact spaces which are not compact. | For example, in an infinite discrete space \( X \), the filter consisting of those sets which contain a given point \( x \in \mathbf{X} \) and have a finite complement has \( x \) as its only cluster point but does not converge to \( x \) . Since any mapping \( f \) of \( \mathrm{X} \) into a Hausdorff space \( {\mathr... | Yes |
Proposition 15. If \( \mathrm{X} \) is a locally compact \( \sigma \) -compact space, there is a sequence \( \left( {\mathbf{U}}_{n}\right) \) of relatively compact open subsets of \( \mathbf{X} \) which cover \( \mathbf{X} \), such that \( {\overline{\mathbf{U}}}_{n} \subset {\mathbf{U}}_{n + 1} \) for each \( n \) . | Let \( \mathrm{X} \) be the union of a sequence \( \left( {\mathrm{K}}_{n}\right) \) of compact sets. Let \( {\mathrm{U}}_{1} \) be a relatively compact open neighbourhood of \( {\mathrm{K}}_{1} \) (no. 7, Proposition 10) and define \( {\mathrm{U}}_{n} \) inductively for \( n > \mathrm{I} \) to be a relatively compact ... | Yes |
Corollary 1. With the notation of Proposition 15, every compact subset \( \mathrm{K} \) of \( \mathrm{X} \) is contained in some \( {\mathrm{U}}_{n} \) . | For \( \mathrm{K} \) can be covered by a finite number of the \( {\mathrm{U}}_{k} \), by axiom \( \left( {\mathrm{C}}^{\prime \prime \prime }\right) \) . | No |
Corollary 2. Let \( \mathrm{X} \) be a locally compact space and let \( {\mathrm{X}}^{\prime } \) be the compact space obtained by adjoining a point at infinity \( \omega \) to \( \mathrm{X} \) (no. 8). Then \( \mathrm{X} \) is \( \sigma \) -compact if and only if the point \( \omega \) has a countable fundamental syst... | If \( \mathrm{X} \) is \( \sigma \) -compact we can construct a sequence of subsets \( {\mathrm{U}}_{n} \) of \( \mathrm{X} \) as in Proposition 15, and the neighbourhoods \( {}^{ \cdot }{\mathrm{X}}^{\prime } - {\overline{\mathrm{U}}}_{n} \) of \( \omega \) in \( {\mathrm{X}}^{\prime } \) form a fundamental system of ... | Yes |
Proposition 16. Every closed subspace \( \mathbf{F} \) of a paracompact space \( \mathbf{X} \) is para-compact. | Certainly \( \mathrm{F} \) is Hausdorff. On the other hand, if \( \left( {\mathrm{V}}_{\mathrm{t}}\right) \) is an open covering in the subspace \( \mathrm{F} \), then each \( {\mathrm{V}}_{\mathrm{t}} \) is of the form \( {\mathrm{V}}_{\mathrm{t}} = {\mathrm{U}}_{\mathrm{t}} \cap \mathrm{F} \), where \( {\mathrm{U}}_{... | Yes |
Proposition 17. The product of a paracompact space and a compact space is paracompact. | Let \( \mathrm{X} \) be a paracompact space, \( \mathrm{Y} \) a compact space, \( \Re \) an open covering of \( \mathrm{X} \times \mathrm{Y} \) . For each \( \left( {x, y}\right) \in \mathrm{X} \times \mathrm{Y} \) there is an open neighbourhood \( \mathrm{V}\left( {x, y}\right) \) of \( x \) in \( \mathrm{X} \) and an... | Yes |
Proposition 18. The sum (§ 2, no. 4, Example 3) of a family \( {\left( {\mathrm{X}}_{\mathrm{t}}\right) }_{\mathrm{t} \in \mathrm{I}} \) of paracompact spaces is paracompact. | Let \( X \) be the sum of the \( {X}_{t} \), and let \( {\left( {V}_{\lambda }\right) }_{\lambda \in L} \) be an open covering of \( X \) . The covering formed by the open sets \( {\mathrm{X}}_{\mathrm{t}} \cap {\mathrm{Y}}_{\lambda } \) is finer than \( \left( {\mathrm{V}}_{\lambda }\right) \) . For each \( \iota \in ... | No |
Proposition 2. Let \( f : \mathrm{X} \rightarrow \mathrm{Y} \) be a continuous injection. Then the following three statements are equivalent :\na) \( f \) is proper.\nb) \( f \) is closed.\nc) \( f \) is a homeomorphism of \( \mathbf{X} \) onto a closed subset of \( \mathbf{Y} \) . | We have just seen that a) implies b). Since the equivalence relation \( f\left( x\right) = f\left( {x}^{\prime }\right) \) is the equality relation, the quotient space of \( \mathrm{X} \) with respect to this relation can be identified with \( \mathrm{X} \) ; hence b) implies c) by reason of \( §5 \), no. 2, Propositio... | Yes |
a) If \( f \) is proper, so is \( {f}_{\mathbf{T}} \) . | Let \( Z \) be a topological space. If \( T \) is any subset of \( Y \), we have\n\n\[ \n{f}_{\mathbf{T}} \times {\iota }_{\mathbf{Z}} = {\left( f \times {\iota }_{\mathbf{Z}}\right) }_{\mathbf{T} \times \mathbf{Z}} \]\n\nif \( f \) is proper, then \( f \times {\iota }_{\mathbf{Z}} \) is closed, hence so is \( {\left( ... | Yes |
Proposition 4. Let \( \mathrm{I} \) be a finite set and for each \( i \in \mathrm{I} \) let \( {f}_{i} : {\mathrm{X}}_{i} \rightarrow {\mathrm{Y}}_{i} \) be a continuous mapping. Let \( \mathrm{X} = \mathop{\prod }\limits_{{i \in \mathbf{I}}}{\mathrm{X}}_{i},\mathrm{Y} = \mathop{\prod }\limits_{{i \in \mathbf{I}}}{\mat... | By induction it is enough to consider the case where \( I = \{ 1,2\} \) .\n\na) Suppose that \( {f}_{1},{f}_{2} \) are proper, and let \( \mathbf{Z} \) be a topological space; \( {f}_{1} \times {f}_{2} \times {\iota }_{Z} \) is the composition of \( {\iota }_{{Y}_{1}} \times {f}_{2} \times {\iota }_{Z} \) and\n\n\[ {f}... | Yes |
Proposition 5. Let \( f : \mathrm{X} \rightarrow {\mathrm{X}}^{\prime } \) and \( g : {\mathrm{X}}^{\prime } \rightarrow {\mathrm{X}}^{\prime \prime } \) be two continuous mappings.\na) If \( f \) and \( g \) are proper, then \( g \circ f \) is proper. | Let \( Z \) be a topological space. We have\n\n\[ \left( {g \circ f}\right) \times {\mathfrak{\iota }}_{\mathbf{Z}} = \left( {g \times {\mathfrak{\iota }}_{\mathbf{Z}}}\right) \circ \left( {f \times {\mathfrak{\iota }}_{\mathbf{Z}}}\right) \]\n\nif \( f \) and \( g \) are proper, then \( f \times {\mathfrak{r}}_{\mathb... | No |
Corollary 1. If \( f : \mathrm{X} \rightarrow \mathrm{Y} \) is a proper mapping, then the restriction of \( f \) to a closed subset \( \mathbf{F} \) of \( \mathbf{X} \) is a proper mapping of \( \mathbf{F} \) into \( \mathrm{Y} \) . | For this restriction is the composition \( f \circ j \), where \( j : \mathrm{F} \rightarrow \mathrm{X} \) is the canonical injection, which is proper by Proposition 2. | Yes |
Corollary 2. Let \( f : \mathrm{X} \rightarrow \mathrm{Y} \) be a proper mapping, where \( \mathrm{X} \) is Hausdorff. Then the subspace \( f\left( \mathbf{X}\right) \) of \( \mathbf{Y} \) is Hausdorff. | By reason of Proposition 5 c) we need only consider the case where \( f\left( \mathrm{X}\right) = \mathrm{Y} \) . Then the diagonal of \( \mathrm{Y} \times \mathrm{Y} \) is the image under \( f \times f \) of the diagonal of \( \mathrm{X} \), which is closed \( \left( {§8\text{, no. I, Proposition I}}\right) ;f \times ... | Yes |
Corollary 3. Let \( \mathrm{I} \) be a finite set and for each \( i \in \mathrm{I} \), let \( {f}_{i} : \mathrm{X} \rightarrow {\mathrm{Y}}_{i} \) be a proper mapping. If \( \mathrm{X} \) is Hausdorff, then the mapping \( x \rightarrow \left( {{f}_{i}\left( x\right) }\right) \) of \( \mathrm{X} \) into \( \mathop{\prod... | This mapping is the composition of the product mapping \( \left( {x}_{i}\right) \rightarrow \left( {{f}_{i}\left( {x}_{i}\right) }\right) \) of \( {\mathrm{X}}^{\mathrm{I}} \) into \( \mathop{\prod }\limits_{i}{\mathrm{Y}}_{i} \) and the diagonal mapping of \( \mathrm{X} \) into \( {\mathrm{X}}^{\mathrm{I}} \) ; since ... | Yes |
Corollary 4. Let \( \\mathrm{X} \) and \( \\mathrm{Y} \) be two topological spaces, \( f : \\mathrm{X} \\rightarrow \\mathrm{Y} \) a continuous mapping, \( \\mathbf{R} \) the equivalence relation \( f\\left( x\\right) = f\\left( y\\right) \) on \( \\mathbf{X} \), and\n\n\[ \n\\mathrm{X}\\overset{p}{ \\rightarrow }\\mat... | The conditions are sufficient by virtue of Proposition 5 a) and Proposition 2. Conversely, if \( f \) is proper, then \( f \) is closed; hence \( f\\left( \\mathrm{X}\\right) \) is closed in \( \\mathrm{Y} \) and \( h \) is a homeomorphism (§ 5, no. 2, Proposition 3); also \( h \\circ p \) is proper by Proposition \( 5... | Yes |
Lemma 1. Let \( \mathrm{X} \) be a topological space such that the constant mapping \( \mathrm{X} \rightarrow \mathrm{P} \) is proper. Then \( \mathrm{X} \) is quasi-compact. | (We shall see a little later on (Theorem 1, Corollary 1) that this property characterizes quasi-compact spaces.)\n\nWe may restrict ourselves to the case where \( \mathbf{X} \) is not empty. Let \( \mathfrak{F} \) be a filter on \( \mathrm{X} \), and \( {\mathrm{X}}^{\prime } = \mathrm{X} \cup \{ \omega \} \) the topol... | No |
Lemma 2. If \( {\left( {f}_{i}\right) }_{i \in \mathbf{I}} \) is a family of continuous mappings \( {f}_{i} : {\mathrm{X}}_{i} \rightarrow {\mathrm{Y}}_{i} \) each of which satisfies condition \( \mathrm{d} \) of Theorem 1, then the product mapping\n\n\[ f : \left( {x}_{t}\right) \rightarrow \left( {{f}_{t}\left( {x}_{... | Let \( \mathfrak{u} \) be an ultrafilter on \( \mathrm{X} = \mathop{\prod }\limits_{i}{\mathrm{X}}_{i} \), and let \( y = \left( {y}_{i}\right) \) be a point of \( \mathrm{Y} = \mathop{\prod }\limits_{i}{\mathrm{Y}}_{i} \) such that \( f\left( \mathfrak{U}\right) \) converges to \( y \) . This means that each of the ul... | No |
Corollary 1. A topological space \( \mathrm{X} \) is quasi-compact if and only if the mapping \( \mathrm{X} \rightarrow \mathrm{P} \) is proper. | Apply a) \( \Leftrightarrow \) b) to \( \mathrm{X} \rightarrow \mathrm{P} \). | No |
Corollary 3. If \( \left( {f}_{t}\right) \) is a family of proper mappings, then the product mapping \( \left( {x}_{\iota }\right) \rightarrow \left( {{f}_{\iota }\left( {x}_{\iota }\right) }\right) \) is proper. | In view of Theorem 1, this is just Lemma 2 above. | No |
Corollary 4. Let \( \mathrm{X} \) be a Hausdorff space, and let \( {f}_{t} : \mathrm{X} \rightarrow {\mathrm{Y}}_{t} \) be a family of proper mappings. Then the mapping \( f : x \rightarrow \left( {{f}_{\iota }\left( x\right) }\right) \) of \( \mathrm{X} \) into \( \mathop{\prod }\limits_{\iota }{\mathrm{Y}}_{\iota } \... | The proof is the same as in the case of a finite family (no. 1, Proposition 5, Corollary 3), using Corollary 3 above and the fact that the diagonal of \( {\mathrm{X}}^{\mathrm{I}} \) is closed \( \left( {§8\text{, no. I, Proposition I}}\right) \) . | No |
Corollary 5. If \( \mathrm{X} \) is any quasi-compact space and \( \mathrm{Y} \) is any topological space, then the projection \( {\mathrm{{pr}}}_{2} : \mathrm{X} \times \mathrm{Y} \rightarrow \mathrm{Y} \) is proper. | For we may identify \( \mathrm{Y} \) with \( \mathrm{P} \times \mathrm{Y} \) and \( {\mathrm{{pr}}}_{2} \) with the product of \( \mathrm{X} \rightarrow \mathrm{P} \) and \( {\iota }_{Y} \), both of which are proper mappings. | No |
Proposition 6. Let \( f : \mathrm{X} \rightarrow \mathrm{Y} \) be a proper mapping, and let \( \mathrm{K} \) be a quasi-compact subset of \( \mathrm{Y} \). Then \( {\bar{f}}^{1}\left( \mathrm{\;K}\right) \) is quasi-compact. | By Proposition 3 of no. 1, the mapping \( {f}_{\mathbf{K}} : {\overrightarrow{f}}^{1}\left( \mathrm{\;K}\right) \rightarrow \mathrm{K} \) is proper. Since \( \mathrm{K} \rightarrow \mathrm{P} \) is a proper mapping (Theorem 1, Corollary 1) it follows from no. 1, Proposition 5 a) that the composition \( {\bar{f}}^{1}\le... | Yes |
Proposition 7. Let \( f \) be a continuous mapping of a Hausdorff space \( \mathbf{X} \) into a locally compact space \( \mathrm{Y} \) . Then \( f \) is proper if and only if the inverse image under \( f \) of every compact subset of \( \mathrm{Y} \) is compact. Further, if \( f \) is proper then \( \mathrm{X} \) is lo... | If \( f \) is proper and \( \mathrm{K} \) is a compact subset of \( \mathrm{Y} \), then \( {\bar{f}}^{1}\left( \mathrm{\;K}\right) \) is compact by Proposition 6 of no. 2. Conversely, if this condition is satisfied, let \( \left( {U}_{\alpha }\right) \) be a covering of \( Y \) by relatively compact open sets. Then the... | Yes |
Proposition 8. Let \( \mathrm{X} \) be a compact space, \( \mathrm{R} \) an equivalence relation on \( \mathrm{X} \) , C the graph of \( \mathrm{R} \) in \( \mathrm{X} \times \mathrm{X}, f \) the canonical mapping \( \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) . Then the following conditions are equivalent:\n\na) \... | R is closed if and only if \( f \) is closed; hence b) implies c) by reason of Theorem Ib) of no. 2. That c) implies d) is a particular case of no. I, Proposition 5, Corollary 2. d) implies a) for any topological space \( \mathrm{X} \) (§ 8, no. 3, Proposition 8). It remains to show that a) implies b). If \( \mathrm{F}... | Yes |
Proposition 9. Let \( \mathrm{X} \) be a locally compact space, \( \mathrm{R} \) an equivalence relation on \( \mathrm{X} \), C the graph of \( \mathrm{R} \) in \( \mathrm{X} \times \mathrm{X}, f \) the canonical mapping \( \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) ; let \( {\mathrm{X}}^{\prime } \) be the compac... | a) \( \Rightarrow \) b): Since \( \mathrm{X}/\mathrm{R} = f\left( \mathrm{X}\right) \) and \( f \) is proper, \( \mathrm{X}/\mathrm{R} \) is Hausdorff (no. 1, Proposition 5, Corollary 2); hence the image under \( f \) of every compact subset \( \mathrm{K} \) of \( \mathrm{X} \) is compact \( \left( {§9,\text{no. 4, The... | Yes |
Proposition 10. Let \( \mathrm{X} \) be a locally compact space, \( \mathrm{R} \) an open Hausdorff equivalence relation on \( \mathrm{X} \), and let \( f : \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) be the canonical mapping. Then \( \mathrm{X}/\mathrm{R} \) is locally compact, and if \( {\mathrm{K}}^{\prime } \) ... | The first assertion is a consequence of the facts that each \( x \in \mathrm{X} \) has a compact neighbourhood \( \mathrm{V} \) and that \( f\left( \mathrm{\;V}\right) \) is a compact neighbourhood of \( f\left( x\right) \) (§ 5, no. 3, Proposition 5 and § 9, no. 4, Theorem 2, Corollary 1). For each \( y \in {\mathrm{K... | Yes |
Proposition 2. The union of a family of connected sets whose intersection is non-empty is connected. | Let \( {\left( {\mathbf{A}}_{\iota }\right) }_{\iota \in \mathbf{I}} \) be a family of connected subsets of \( \mathbf{X} \), all of which contain the same point \( x \) ; we have to show that\n\n\[ A = \mathop{\bigcup }\limits_{i}{A}_{t} \]\n\n is connected. If not, there are two open sets \( \mathbf{B} \) and \( \mat... | Yes |
Proposition 3. Let \( \mathrm{A} \) be a subset of a topological space \( \mathrm{X} \). If \( \mathrm{B} \) is a connected subset of \( \mathrm{X} \) which meets both \( \mathrm{A} \) and \( \complement \mathrm{A} \), then \( \mathrm{B} \) meets the frontier of \( \mathbf{A} \). | For otherwise the intersections of \( \mathbf{B} \) with the interior and exterior of \( \mathbf{A} \) would be two open subsets of \( \mathbf{B} \) which form a partition of \( \mathbf{B} \), and \( \mathbf{B} \) would not be connected. | Yes |
Proposition 4. Let \( \mathrm{A} \) be a connected subset of a topological space \( \mathrm{X} \), and let \( f \) be a continuous mapping of \( \mathrm{X} \) into a topological space \( {\mathrm{X}}^{\prime } \) . Then \( f\left( \mathrm{\;A}\right) \) is connected. | Suppose \( f\left( \mathrm{\;A}\right) \) is not connected. Then there exist two sets \( {\mathrm{M}}^{\prime },{\mathrm{N}}^{\prime } \) which are open in \( f\left( \mathrm{\;A}\right) \) and which form a partition of \( f\left( \mathrm{\;A}\right) \) ; hence \( \mathrm{A} \cap {\bar{f}}^{1}\left( {\mathrm{M}}^{\prim... | Yes |
Proposition 5. For a topological space \( \mathbf{X} \) to be not connected it is necessary and sufficient that there exists a surjective continuous mapping of \( \mathbf{X} \) onto a discrete space containing more than one point. | The condition is sufficient by Proposition 4. Conversely, if \( \mathrm{X} \) is not connected, there exist two non-empty disjoint open subsets A, B whose union is \( \mathrm{X} \), and the mapping \( f \) of \( \mathrm{X} \) onto a discrete space of two elements \( \{ a, b\} \), defined by \( f\left( \mathrm{\;A}\righ... | Yes |
Proposition 6. Every quotient space of a connected space is connected. | This is an immediate consequence of Proposition 4 of no. 2. | No |
Proposition 7. Let \( \mathrm{X} \) be a topological space and \( \mathrm{R} \) an equivalence relation on \( \mathrm{X} \) . If the quotient space \( \mathrm{X}/\mathrm{R} \) is connected, and if each equivalence class mod \( \mathrm{R} \) is connected, then \( \mathrm{X} \) is connected. | Suppose \( \mathrm{X} \) is not connected. Then there is a partition of \( \mathrm{X} \) into two open sets A, B. The sets A, B are saturated with respect to R ; for if \( x \in \mathrm{A} \) then the equivalence class \( \mathrm{M} \) of \( x \) cannot meet \( \mathrm{B} \), otherwise the sets \( A \cap M, B \cap M \)... | Yes |
Proposition 8. Every product of connected spaces is connected. Conversely, if a product of non-empty spaces is connected, then each of the factors is connected. | Let \( \mathrm{X} = \mathop{\prod }\limits_{{i \in \mathrm{I}}}{\mathrm{X}}_{i} \) be a product of topological spaces. If the \( {\mathrm{X}}_{i} \) are non-empty, we have \( {X}_{t} = {\operatorname{pr}}_{t}X \) for each \( \iota \in I \) ; hence if \( X \) is connected so are the \( {X}_{i} \) (no. 2, Proposition 4).... | Yes |
Proposition 10. In a product space \( \mathbf{X} = \mathop{\prod }\limits_{{i \in \mathbf{J}}}{\mathbf{X}}_{i} \), the component of \( x = \left( {x}_{i}\right) \) in \( \mathrm{X} \) is the product of the components of \( {x}_{i} \) in the factors \( {\mathrm{X}}_{i} \) . | This product set is connected (no. 4, Proposition 8). Conversely, if A is a connected subset of \( \mathrm{X} \) which contains \( x \), then \( {\operatorname{pr}}_{\mathrm{t}}\left( \mathrm{A}\right) \) is a connected set (no. 2, Proposition 4) which contains \( x \) ; since \( \mathrm{A} \subset \mathop{\coprod }\li... | No |
Proposition 11. A necessary and sufficient condition for a space \( \mathrm{X} \) to be locally connected is that every component of an open set in \( \mathrm{X} \) is open in \( \mathrm{X} \) . | The condition is sufficient, since the component of \( x \) relative to an open neighbourhood of \( x \) is then a neighbourhood of \( x \) in \( \mathrm{X} \) .\n\nConversely, let \( \mathrm{A} \) be an open subset of a locally connected space \( \mathrm{X} \) , let \( \mathbf{B} \) be a component of \( \mathbf{A} \),... | Yes |
Proposition 12. Every quotient space of a locally connected space is locally connected. | Let \( \mathrm{X} \) be a locally connected space, \( \mathrm{R} \) an equivalence relation on \( \mathrm{X} \) , \( \varphi : \mathrm{X} \rightarrow \mathrm{X}/\mathrm{R} \) the canonical mapping. Let \( \mathrm{A} \) be an open subset of\n\n\( \mathrm{X}/\mathrm{R} \) and \( \mathrm{C} \) a component of \( \mathrm{A}... | Yes |
Proposition 13. a) Let \( {\left( {\mathrm{X}}_{\iota }\right) }_{\iota \in \mathbf{I}} \) be a family of locally connected spaces such that \( {\mathrm{X}}_{\mathrm{t}} \) is connected for all but a finite number of indices \( \mathfrak{t} \in \mathrm{I} \) . Then the product space \( \mathbf{X} = \mathop{\prod }\limi... | a) Let \( \mathbf{J} \) be the finite subset of \( \mathbf{I} \) such that \( {\mathbf{X}}_{i} \) is not connected if and only if \( i \in J \) . Let\n\n\[ U = \mathop{\prod }\limits_{{\iota \in I}}{U}_{\iota } \]\n\nbe an elementary set containing a point \( x = \left( {x}_{t}\right) \) of \( \mathrm{X} \) and let \( ... | Yes |
Corollary 1. Let \( Y \) be a regular space whose topology has a countable base (*). Let \( \mathrm{X} \) be a connected and locally connected space, and let \( p : \mathrm{X} \rightarrow \mathrm{Y} \) be a continuous mapping with the following property: for each \( x \in \mathrm{X} \) there is a closed neighbourhood \... | First, the hypotheses imply that \( \mathbf{X} \) is regular \( \left( {§8,\text{no. 4, Proposition 13}}\right) \) . Let us show that the conditions of Theorem 1 are satisfied if we take \( \mathfrak{B} \) to be the set of all closed subsets \( V \) of \( X \) such that the restriction of \( p \) to \( \mathrm{V} \) is... | Yes |
Corollary 2. Let \( \mathrm{X} \) be locally compact, connected and locally connected, and suppose each point of \( \mathbf{X} \) has a neighbourhood which has a countable base. Let \( \mathrm{Y} \) be a Hausdorff space whose topology has a countable base, and let \( p \) : \( \mathrm{X} \rightarrow \mathrm{Y} \) be a ... | For each \( x \in \mathrm{X} \), let \( {\mathrm{V}}_{x} \) be a compact neighbourhood of \( x \) in \( \mathrm{X} \) which has a countable base. It follows from \( §9 \), no. 4, Theorem 2, Corollary 2, that the set \( \mathfrak{B} \) of the \( {\mathrm{V}}_{x} \) satisfies the conditions of Theorem 1, and we complete ... | No |
Proposition 1. Let \( \mathrm{X} \) be a set endowed with a uniform structure \( \mathrm{u} \), and for each \( x \in \mathrm{X} \) let \( \mathfrak{B}\left( x\right) \) be the set of subsets \( \mathrm{V}\left( x\right) \) of \( \mathrm{X}\left( *\right) \), where \( \mathrm{V} \) runs through the set of entourages of... | We have to show that the \( \mathfrak{V}\left( x\right) \) satisfy conditions \( \left( {\mathrm{V}}_{\mathrm{I}}\right) ,\left( {\mathrm{V}}_{\mathrm{{II}}}\right) ,\left( {\mathrm{V}}_{\mathrm{{III}}}\right) \) and \( \left( {\mathrm{V}}_{\mathrm{{IV}}}\right) \) of Chapter \( \mathrm{I},§\mathrm{I} \), no. 2. That t... | Yes |
Proposition 2. Let \( \mathrm{X} \) be a uniform space. For every symmetric entourage \( \mathrm{V} \) of \( \mathrm{X} \) and every subset \( \mathrm{M} \) of \( \mathrm{X} \times \mathrm{X} \), VMV is a neighbourhood of \( \mathrm{M} \) in the product space \( \mathbf{X} \times \mathbf{X} \), and the closure of \( \m... | Let \( \mathrm{V} \) be a symmetric entourage of \( \mathrm{X} \) . The relation \( \left( {x, y}\right) \in \mathrm{{VMV}} \) means that there is an element \( \left( {p, q}\right) \) of \( \mathbf{M} \) such that \( \left( {x, p}\right) \in \mathrm{V} \) and \( \left( {q, y}\right) \in \mathrm{V} \) : in other words ... | Yes |
Corollary 2. The interiors (resp. the closures) of the entourages of \( \mathrm{X} \) in \( \mathrm{X} \times \mathrm{X} \) form a fundamental system of entourages of \( \mathrm{X} \) . | If \( \mathrm{V} \) is any entourage of \( \mathrm{X} \), there is a symmetric entourage \( \mathrm{W} \) such that \( \mathrm{W} \subset \mathrm{V} \) ; since \( \mathrm{W} \) is a neighbourhood of \( \mathrm{W} \) (Proposition 2), the interior of \( \mathbf{V} \) in \( \mathbf{X} \times \mathbf{X} \) contains \( \mat... | Yes |
Corollary 3. Every uniform space satisfies axiom \( \left( {\mathrm{O}}_{\mathrm{{III}}}\right) \) . | If \( x \) is any point of \( \mathrm{X} \) and \( \mathrm{V} \) runs through the entourages of \( \mathrm{X} \) which are closed in \( \mathrm{X} \times \mathrm{X} \), then the sets \( \mathrm{V}\left( x\right) \) form a fundamental system of neighbourhoods of \( x \) in \( \mathrm{X} \) by Corollary 2, and they are c... | Yes |
Proposition 3. A uniform space \( \mathrm{X} \) is Hausdorff if and only if the intersection of all the entourages of its uniform structure is the diagonal \( \Delta \) of \( \mathrm{X} \times \mathrm{X} \). | We have seen that the closed entourages form a fundamental system of entourages (Proposition 2, Corollary 2); if their intersection is \( \Delta \), then \( \Delta \) is closed in \( \mathrm{X} \times \mathrm{X} \) and consequently \( \mathrm{X} \) is Hausdorff (Chapter I, § 8, no. 1, Proposition 1). Conversely, if X i... | Yes |
Proposition 1. Every uniformly continuous mapping is continuous. | This is an immediate consequence of the definitions. | 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 uniform spaces, and for each \( \iota \in \mathrm{I} \) let \( {f}_{\iota } \) be a mapping of \( \mathrm{X} \) into \( {\mathrm{Y}}_{\iota } \) . For each \( \iota \in \mathrm{I... | It is immediately seen that \( \mathfrak{B} \) satisfies axioms \( \left( {\mathbf{B}}_{\mathbf{1}}\right) \) Tand \( \left( {\mathbf{U}}_{\mathbf{1}}^{\prime }\right) \) . If \( {\mathrm{W}}_{t} = {\bar{g}}_{t}\left( {\mathrm{\;V}}_{t}\right) \), then \( {\overline{\mathrm{W}}}_{t} = {\bar{g}}_{t}\left( {\overline{\ma... | Yes |
Proposition 6. Let \( \mathrm{A} \) be a dense subset of a uniform space \( \mathrm{X} \). Then the closures, in \( \mathrm{X} \times \mathrm{X} \), of the entourages of the uniform subspace \( \mathrm{A} \) form a fundamental system of entourages of \( \mathbf{X} \). | \( \mathrm{A} \times \mathrm{A} \) is dense in \( \mathrm{X} \times \mathrm{X} \) (Chapter I,§ 4, no. 3, Proposition 7). Let \( \mathrm{V} \) be an open entourage of \( A \); it is the intersection of \( A \times A \) with an open entourage \( U \) of \( X \). We have \( U \subset \bar{V} \) (Chapter \( I,§I \), no. 6,... | Yes |
Proposition 7. Let \( f = \left( {f}_{i}\right) \) be a mapping of a uniform space \( \mathrm{Y} \) into a product uniform space \( \mathrm{X} = \mathop{\prod }\limits_{{i \in I}}{\mathrm{X}}_{i} \) . Then \( f \) is uniformly continuous if and only if each \( {f}_{t} \) is uniformly continuous. | Since \( {f}_{1} = {\operatorname{pr}}_{1} \circ f \), this is a particular case of Proposition 4 of no. 3 . | No |
Proposition 10. Let \( \mathrm{I} \) be a directed set, let \( \left( {{\mathrm{X}}_{\alpha },{f}_{\alpha \beta }}\right) \) be an inverse system of uniform spaces indexed by \( \mathbf{I} \), and let \( \mathbf{J} \) be a cofinal subset of \( \mathbf{I} \) . For each \( \alpha \in \mathbf{I} \) let \( {f}_{\alpha } \)... | We leave the proof to the reader; it is a straightforward adaptation of the proof of Proposition 9 of Chapter I,§ 4, no. 4. | No |
Proposition 1. In a uniform space \( \mathrm{X} \), if two sets \( \mathrm{A} \) and \( \mathrm{B} \) are \( \mathrm{V} \) -small and intersect, then their union \( \mathrm{A} \cup \mathrm{B} \) is \( \widehat{\mathrm{V}} \) -small. | Let \( x \) and \( y \) be any two points of \( \mathrm{A} \cup \mathrm{B} \), and let \( \mathrm{Z} \Subset \mathrm{A} \cap \mathrm{B} \) . Then \( \left( {x, z}\right) \in \mathbf{V} \) and \( \left( {z, y}\right) \in \mathbf{V} \), so that \( \left( {x, y}\right) \in \overset{2}{\mathbf{V}} \) . | Yes |
Proposition 2. On a uniform space \( \mathrm{X} \) every convergent filter is a Cauchy filter. | If \( x \) is any point of \( \mathrm{X} \) and \( \mathrm{V} \) is any symmetric entourage of \( \mathrm{X} \), then the neighbourhood \( \mathrm{V}\left( x\right) \) of \( x \) is \( \mathrm{V} \) -small. If \( \mathfrak{F} \) is a filter which converges to \( x \), there is a set of \( \mathfrak{F} \) contained in \... | Yes |
Proposition 3. Let \( f : \mathrm{X} \rightarrow {\mathrm{X}}^{\prime } \) be a uniformly continuous mapping. Then the image under \( f \) of a Cauchy filter base on \( \mathbf{X} \) is a Cauchy filter base on \( {\mathbf{X}}^{\prime } \). | Let \( g = f \times f \) . If \( {\mathrm{V}}^{\prime } \) is an entourage of \( {\mathrm{X}}^{\prime } \), then \( {}_{g}^{-1}\left( {\mathrm{\;V}}^{\prime }\right) \) is an entourage of \( \mathrm{X} \), and the image under \( f \) of a \( {}^{-1}\left( {\mathrm{\;V}}^{\prime }\right) \) -small set is \( {\mathrm{V}}... | 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 uniform spaces, and for each \( v \in \mathbf{I} \) let \( {f}_{i} \) be a mapping of \( \mathrm{X} \) into \( {\mathrm{Y}}_{i} \) . Let \( \mathrm{X} \) carry the coarsest unifo... | The condition is necessary by Proposition 3. Conversely, suppose that it is satisfied, and let \( U\left( {{V}_{{t}_{1}},\ldots ,{V}_{{t}_{n}}}\right) \) be an entourage of the uniformity \( \mathcal{U} \) [§ 2, no. 3, formula (I)]. By hypothesis, for each index \( k \) there is a set \( {\mathbf{M}}_{k} \in \mathfrak{... | Yes |
Proposition 5. Let \( \mathrm{X} \) be a uniform space. For each Cauchy filter \( \mathfrak{F} \) on \( \mathrm{X} \) there is a unique minimal Cauchy filter \( {\mathfrak{F}}_{0} \) coarser than \( \mathfrak{F} \) . If \( \mathfrak{B} \) is a base of \( \mathfrak{F} \) and \( \mathfrak{S} \) is a fundamental system of... | If \( \mathrm{M},{\mathrm{M}}^{\prime } \) are in \( \mathfrak{B} \) and \( \mathrm{V},{\mathrm{V}}^{\prime } \) are in \( \mathfrak{S} \), then there is a set \( {\mathbf{M}}^{\prime \prime } \in \mathfrak{B} \) (resp. \( {\mathrm{V}}^{\prime \prime } \in \mathfrak{S} \) ) such that \( {\mathbf{M}}^{\prime \prime } \s... | Yes |
Corollary 3. Every Cauchy filter, which is coarser than a filter converging to a point \( x \), also converges to \( x \) . | This is a consequence of Corollary 2. | No |
Corollary 4. If \( \mathfrak{F} \) is a minimal Cauchy filter, then every set of \( \mathfrak{F} \) has a nonempty interior which also belongs to \( \mathfrak{F} \) (in other words, \( \mathfrak{F} \) has a base consisting of open sets). | Let \( \mathrm{V} \) be any entourage of \( \mathrm{X} \) ; then there is an open entourage \( \mathrm{U} \subset \mathrm{V} \) (§ 1, no. 2, Corollary 2 to Proposition 2). For each subset \( M \) of \( X, U\left( M\right) \) is open and contained in \( \mathrm{V}\left( \mathrm{M}\right) \) ; hence the result, in view o... | No |
Proposition 6. Let \( \mathfrak{F} \) be a filter on a set \( \mathrm{X} \), and let \( f \) be a mapping of \( \mathrm{X} \) into a complete uniform space \( {\mathrm{X}}^{\prime } \) . Then \( f \) has a limit with respect to \( \mathfrak{F} \) if and only if the image of \( \mathfrak{F} \) under \( f \) is a Cauchy ... | This criterion shows the importance of complete spaces in all questions involving the notion of limit: if a function takes its values in a complete space we can prove the existence of a limit without knowing in advance the value of the limit; this would be impossible if the definition of limit were the only criterion o... | No |
Proposition 7. Let \( {\mathcal{U}}_{1},{\mathcal{U}}_{2} \) be two uniformities on a set \( \mathrm{X} \), and let \( {\mathcal{C}}_{1},{\mathcal{C}}_{2} \) be the topologies induced by these uniformities respectively. Suppose that \( {u}_{1} \) is finer than \( {\mathcal{U}}_{2} \), and that there is a fundamental sy... | The conditions are clearly necessary, because \( {\mathcal{C}}_{2} \) is coarser than \( {\mathcal{C}}_{1} \) . Conversely, suppose that the conditions are satisfied, and let \( x \) be a limit point of \( \mathfrak{F} \) with respect to \( {\mathbb{C}}_{2} \) ; we shall show that \( x \) is a limit of \( \mathfrak{F} ... | Yes |
Proposition 8. Every closed subspace of a complete space is complete. Every complete subspace of a Hausdorff uniform space (complete or not) is closed. | Let \( \mathrm{X} \) be a complete space and let \( \mathrm{A} \) be a closed subspace of \( \mathrm{X} \) . If 8 is a Cauchy filter on A, then it is a Cauchy filter base on \( \mathbf{X} \) (no. 1, Proposition 3) and therefore converges to a point \( x \in \mathrm{X} \) ; but since \( \mathrm{A} \) is closed we have \... | No |
Proposition 9. Let \( \mathrm{X} \) be a uniform space and let \( \mathrm{A} \) be a dense subset of \( \mathrm{X} \) such that every Cauchy filter base on A converges in \( \mathrm{X} \) . Then \( \mathrm{X} \) is complete. | It is enough to show that every minimal Cauchy filter \( \mathfrak{F} \) on \( \mathrm{X} \) is convergent. Since A is dense and since every set of \( \mathfrak{F} \) has a non-empty interior (no. 2, Corollary 4 to Proposition 5), the trace \( {\mathfrak{F}}_{A} \) of \( \mathfrak{F} \) on \( \mathrm{A} \) is a Cauchy ... | Yes |
Proposition 10. Every product of complete uniform spaces is complete. Converse- \( {ly} \), if a product of non-empty uniform spaces is complete, then each of the factors is a complete uniform space. | The first assertion is a consequence of the characterization of Cauchy filters and convergent filters on a product space (no. I, Corollary 2 to Proposition 4 and Chapter I,§ 7, no. 6, Corollary 1 to Proposition 10). Conversely, suppose\n\n\[ \mathrm{X} = \mathop{\prod }\limits_{{i \in I}}{\mathrm{X}}_{i} \]\n\nis compl... | Yes |
Proposition 11. Let \( \mathrm{A} \) be a dense subset of a topological space \( \mathrm{X} \), and let \( f \) be a mapping of \( \mathrm{A} \) into a complete Hausdorff uniform space \( {\mathrm{X}}^{\prime } \) . Then \( f \) can be extended by continuity to \( \mathrm{X} \) if and only if, for each \( x \in \mathrm... | This follows from the theorem of extension by continuity (loc. cit.) because \( {\mathrm{X}}^{\prime } \) is regular ( \( § \) 1, no. 2, Proposition 3) and because on \( {\mathrm{X}}^{\prime } \) convergent filters are the same as Cauchy filters. | No |
Proposition 12. (i) The subspace \( i\left( \mathbf{X}\right) \) is dense in \( \widehat{\mathbf{X}} \) . | (i) and (iii) have been proved in the course of the proof of Theorem 3; | No |
Proposition 13. If \( \mathbf{Y} \) is a complete Hausdorff uniform space and \( \mathbf{X} \) a dense subspace of \( \mathrm{Y} \), then the canonical injection \( \mathrm{X} \rightarrow \mathrm{Y} \) extends to an isomorphism of \( \widehat{\mathrm{X}} \) onto \( \mathrm{Y} \) . | For every uniformly continuous mapping of \( \mathrm{X} \) into a complete Hausdorff uniform space \( Z \) extends uniquely to a uniformly continuous mapping of \( \mathrm{Y} \) into \( \mathrm{Z} \) by Theorem 2 of no. 6. | No |
Proposition 14. Let \( \mathrm{X} \) be a complete Hausdorff uniform space, \( \mathcal{U} \) its uniformity, and let \( \mathrm{Z} \) be a dense subspace of \( \mathrm{X} \) . If \( {\mathcal{U}}^{\prime } \) is a uniformity on \( \mathrm{X} \) which is coarser than \( \mathcal{U} \) and which induces the same uniform... | Let \( {\mathrm{X}}^{\prime } \) denote the set \( \mathrm{X} \) with the uniformity \( {\mathcal{U}}^{\prime } \) . The composition of the canonical mapping \( {\mathrm{X}}^{\prime } \rightarrow {\widehat{\mathrm{X}}}^{\prime } \) and the identity mapping \( \mathrm{X} \rightarrow {\mathrm{X}}^{\prime } \) is a unifor... | Yes |
Proposition 15. Let \( \\mathrm{X} \) and \( {\\mathrm{X}}^{\\prime } \) be two uniform spaces. For each uniformly continuous mapping \( f : \\mathrm{X} \\rightarrow {\\mathrm{X}}^{\\prime } \) there is a unique uniformly continuous mapping \( \\widehat{f} : \\widehat{\\mathrm{X}} \\rightarrow {\\widehat{\\mathrm{X}}}^... | Apply Theorem 3 to the function \( {i}^{\\prime } \\circ f : \\mathrm{X} \\rightarrow {\\widehat{\\mathrm{X}}}^{\\prime } \). | Yes |
Proposition 16. Let \( \mathrm{X} \) be a uniform space and \( i \) the canonical mapping of \( \mathrm{X} \) into its Hausdorff completion \( \widehat{\mathrm{X}} \) . For each uniformly continuous mapping \( f \) of \( \mathrm{X} \) into a Hausdorff uniform space \( \mathrm{Y} \), there is a unique uniformly continuo... | We may identify \( Y \) with a subspace of its completion \( \widehat{Y} \) (no. 7, Corollary to Proposition 12), and \( f \) can then be considered as a uniformly continuous mapping of \( \mathrm{X} \) into \( \widehat{\mathrm{Y}} \) . By virtue of Theorem \( 3, f \) is then of the form \( f = g \circ i \), where \( g... | Yes |
Proposition 17. Let \( \mathbf{X} \) be a uniform space, \( i\left( \mathbf{X}\right) \) its associated Hausdorff space, and let \( f \) be a mapping of \( \mathrm{X} \) onto a Hausdorff uniform space \( {\mathrm{X}}^{\prime } \), such that the uniformity of \( \mathbf{X} \) is the inverse image under \( \widetilde{f} ... | By Proposition 16, \( g \) is uniformly continuous; also \( g \) is obviously surjective, and is also injective because the relation \( f\left( x\right) = f\left( y\right) \) implies by definition that \( \left( {x, y}\right) \) belongs to all the entourages of \( \mathrm{X} \), and therefore that \( i\left( x\right) =... | Yes |
Proposition 18. Let \( \mathrm{X} \) be a set, let \( {\left( {\mathrm{Y}}_{\lambda }\right) }_{\lambda \in \mathrm{L}} \) be a family of uniform spaces, and for each \( \lambda \in \mathrm{L} \) let \( {f}_{\lambda } \) be a mapping of \( \mathrm{X} \) into \( {\mathrm{Y}}_{\lambda } \) . Let \( \mathrm{X} \) carry th... | Let \( {\mathrm{X}}^{\prime } \) (resp. \( {\mathrm{Y}}_{\lambda }^{\prime } \) ) be the Hausdorff uniform space associated with \( \mathrm{X} \) (resp. \( {\mathrm{Y}}_{\lambda } \) ), and let \( {f}_{\lambda }^{\prime } : {\mathrm{X}}^{\prime } \rightarrow {\mathrm{Y}}_{\lambda }^{\prime } \) be the uniformly continu... | Yes |
Proposition 1. In a uniform space every subset of a precompact set, every finite union of precompact sets and the closure of every precompact set are precompact. | The first two assertions are immediate consequences of Theorem 3. Let \( \mathrm{X} \) be a uniform space, \( \mathrm{A} \) a precompact subset of \( \mathrm{X} \), and let \( i : \mathrm{X} \rightarrow \widehat{\mathrm{X}} \) be the canonical mapping. \( i\left( \overline{\mathbf{A}}\right) \) is contained in the clos... | Yes |
Proposition 2. Let \( f : \mathrm{X} \rightarrow \mathrm{Y} \) be a uniformly continuous mapping. If \( \mathrm{A} \) is any precompact subset of \( \mathrm{X} \), then \( f\left( \mathrm{\;A}\right) \) is a precompact subset of \( \mathrm{Y} \) . | For if \( i : \mathrm{X} \rightarrow \widehat{\mathrm{X}} \) and \( j : \mathrm{Y} \rightarrow \widehat{\mathrm{Y}} \) are the canonical mappings, we have \( j\left( {f\left( \mathrm{\;A}\right) }\right) = \widehat{f}\left( {i\left( \mathrm{\;A}\right) }\right) \) (§ 3, no. 7, Proposition 15) and hence \( j\left( {f\le... | Yes |
Proposition 3. Let \( \mathrm{X} \) be a set, let \( {\left( {\mathrm{Y}}_{\lambda }\right) }_{\lambda \in \mathrm{L}} \) be a family of uniform spaces, and for each \( \lambda \in \mathrm{L} \) let \( {f}_{\lambda } \) be a mapping of \( \mathrm{X} \) into \( {\mathrm{Y}}_{\lambda } \) . Let \( \mathrm{X} \) carry the... | The condition is necessary by virtue of Proposition 2. Sufficiency follows from the characterization of the Hausdorff completion of \( \mathrm{X} \) given in \( §3 \), no. 9, Proposition 18, and Tychonoff’s theorem (Chapter I,§ 9, Theorem 3, Corollary). | No |
Proposition 4. In a uniform space \( \mathrm{X} \) , let \( \mathrm{A} \) be a compact set and \( \mathrm{B} \) a closed set such that \( \mathrm{A} \cap \mathrm{B} = \varnothing \) . Then there is an entourage \( \mathrm{V} \) of \( \mathrm{X} \) such that \( \mathrm{V}\left( \mathrm{A}\right) \) and \( \mathrm{V}\lef... | If the proposition were false, then none of the sets \( A \cap \overset{2}{V}\left( B\right) \), where \( V \) runs through the set of symmetric entourages of \( \mathbf{X} \), would be empty; hence these sets would form a filter base on \( A \), which would have a cluster point \( {x}_{0} \in \mathrm{A} \) . Hence for... | Yes |
Proposition 6. In a compact space \( \mathrm{X} \), the component of \( x \), the set \( {\mathrm{A}}_{x} \), and the intersection of the neighbourhoods of \( x \) which are both open and closed, are all three identical. | It is enough to show that \( {\mathrm{A}}_{x} \) is connected: for in any uniform space \( \mathrm{X} \) the component of \( x \) is contained in the intersection of the neighbourhoods of \( x \) which are both open and closed, and this intersection is contained in \( {\mathrm{A}}_{x} \) by Proposition 5.\n\nSuppose \(... | Yes |
Proposition 7. Let \( \mathbf{X} \) be a compact space and let \( \mathbf{R} \) be the equivalence relation on \( \mathrm{X} \) whose classes are the components of \( \mathrm{X} \) . Then the quotient space \( \mathrm{X}/\mathrm{R} \) is compact and totally disconnected. | We know from Chapter I, \( § \) II, no. 5, Proposition 9 that \( \mathrm{X}/\mathrm{R} \) is totally disconnected; hence we have to show that X/R is Hausdorff (Chapter I, § 10, no. 4, Proposition 8.) Let A and B be two distinct components of \( \mathrm{X} \) . By Proposition 6 there is a symmetric entourage \( \mathrm{... | Yes |
Proposition 1. Let \( \mathrm{G} \) be a group and let \( \mathfrak{B} \) be a filter on \( \mathrm{G} \) satisfying the axioms \( \left( {\mathrm{{GV}}}_{\mathrm{I}}\right) ,\left( {\mathrm{{GV}}}_{\mathrm{{II}}}\right) \) and \( \left( {\mathrm{{GV}}}_{\mathrm{{III}}}\right) \) . Then there is a unique topology on \(... | If there is a topology with the required properties, then by what has been said above the neighbourhood filter of \( a \) coincides with each of the filters \( a.\mathfrak{V} \) and \( \mathfrak{V}.a \) ; hence the topology is unique, if it exists. Its existence will be established if we show 1) that the filters \( a.\... | Yes |
Proposition 2. A topological group \( \mathrm{G} \) is Hausdorff if and only if the set \( \{ \) e \( \} \) is closed. | Clearly, if \( G \) is Hausdorff, then \( \{ e\} \) is closed. Conversely, if \( \{ e\} \) is closed, then the diagonal \( \Delta \) of \( \mathrm{G} \times \mathrm{G} \) is closed, because it is the inverse image of \( \{ e\} \) under the continuous mapping \( \left( {x, y}\right) \rightarrow x{y}^{-1} \) ; hence (Cha... | Yes |
Proposition 3. Let \( \mathbf{G} \) and \( {\mathbf{G}}^{\prime } \) be two topological groups, and let \( f \) be a homeomorphism of a neighbourhood \( \mathrm{V} \) of the identity element of \( \mathrm{G} \) onto a neighbourhood ’ \( {\mathrm{V}}^{\prime } \) of the identity element of \( {\mathrm{G}}^{\prime } \), ... | For it is not hard to see that if \( \mathrm{W} \) is a neighbourhood of the identity element of \( \mathrm{G} \) such that \( \mathrm{W}.\mathrm{W} \subset \mathrm{V} \), then the restriction of \( f \) to \( \mathrm{W} \) is a local isomorphism of \( \mathrm{G} \) with \( {\mathrm{G}}^{\prime } \). | No |
Proposition 1. The closure \( \overline{\mathbf{H}} \) of a subgroup \( \mathbf{H} \) of a topological group \( \mathbf{G} \) is a subgroup of \( \mathrm{G} \) . If \( \mathrm{H} \) is a normal subgroup of \( \mathrm{G} \), then so is \( \overline{\mathrm{H}} \) . | If \( a, b \in \overline{\mathrm{H}} \) then \( a{b}^{-1} \in \overline{\mathrm{H}} \), because the mapping \( \left( {x, y}\right) \rightarrow x{y}^{-1} \) is continuous on \( \mathrm{G} \times \mathrm{G} \) and transforms \( \mathrm{H} \times \mathrm{H} \) into \( \mathrm{H} \) (Chapter \( \mathrm{I},{\mathrm{S}}_{2}... | Yes |
Proposition 2. If \( \mathrm{G} \) is a Hausdorff topological group, then the closure of a commutative subgroup of \( \mathbf{G} \) is a commutative subgroup of \( \mathbf{G} \) . | By reason of Proposition 1, we may limit ourselves to the case where \( \mathrm{H} \) is dense in G. The continuous functions \( {xy} \) and \( {yx} \) are equal on \( \mathrm{H} \times \mathrm{H} \), and are therefore equal on \( \mathrm{G} \times \mathrm{G} \), by virtue of the principle of extension of identities (C... | Yes |
Proposition 3. Let \( \mathrm{G} \) be a Hausdorff topological group and let \( \mathrm{M} \) be any subset of \( \mathrm{G} \). Then the set \( {\mathrm{M}}^{\prime } \) of elements of \( \mathrm{G} \) which commute with each element of \( \mathbf{M} \) is a closed subgroup of \( \mathbf{G} \). In particular, the cent... | For \( {\mathrm{M}}^{\prime } \) is the intersection of the sets \( {\mathrm{F}}_{m}\left( {m \in \mathrm{M}}\right) \), where \( {\mathrm{F}}_{m} \) is the set of all \( x \in \mathrm{G} \) such that \( {xm} = {mx} \) ; and \( {\mathrm{F}}_{m} \) is closed (Chapter I,§ 8, no. 1, Proposition 2). | Yes |
Proposition 4. Let \( \mathrm{G} \) be a topological group and let \( \mathrm{H} \) be a subgroup of \( \mathrm{G} \) which is locally closed at one point of \( \mathrm{H} \) (Chapter I,§ 3, no. 3, Definition 2). Then \( \mathrm{H} \) is closed in \( \mathrm{G} \) . | By translation, \( \mathrm{H} \) is locally closed at each of its points, i.e. \( \mathrm{H} \) is locally closed in G. Let \( \mathrm{V} \) be a symmetric open neighbourhood of \( e \) in \( \mathrm{G} \) such that \( \mathrm{V} \cap \mathrm{H} \) is closed in \( \mathrm{V} \) . If \( x \in \overline{\mathrm{H}} \), t... | Yes |
A subgroup \( \mathrm{H} \) of a topological group \( \mathrm{G} \) is discrete if and only if \( \mathrm{H} \) has an isolated point. Every discrete subgroup of a Hausdorff group is closed. | If \( \mathrm{H} \) is discrete, every point of \( \mathrm{H} \) is isolated. Conversely if \( \mathrm{H} \) has an isolated point, then by translation every point of \( \mathrm{H} \) is isolated and therefore \( \mathbf{H} \) is discrete. If \( \mathbf{H} \) is discrete and \( \widehat{\mathbf{G}} \) is Hausdorff, the... | Yes |
In a topological group \( \mathrm{G} \), the component \( \mathrm{K} \) of the identity element \( e \) is a closed normal subgroup. The component of any point \( x \in \mathrm{G} \) is the coset \( x \cdot \mathrm{K} = \mathrm{K} \cdot x \). | If \( a \in \mathrm{K} \), then \( {a}^{-1}\mathrm{\;K} \) is connected and contains \( e \) ; hence \( {\mathrm{K}}^{-1}\mathrm{\;K} \subset \mathrm{K} \) , which shows that \( \mathrm{K} \) is a subgroup of \( \mathrm{G} \). This subgroup is invariant under all automorphisms of \( \mathrm{G} \), and in particular und... | No |
Proposition 8. Let \( \mathrm{H} \) be a dense subgroup of a topological group \( \mathrm{G} \), and let \( \mathrm{K} \) be a normal subgroup of \( \mathrm{H} \). Then the closure \( \mathrm{K} \) of \( \mathrm{K} \) in \( \mathrm{G} \) is a normal subgroup of \( \mathrm{G} \). | For the mapping \( \left( {z, x}\right) \rightarrow {zx}{z}^{-1} \) is continuous on \( \mathrm{G} \times \mathrm{G} \), and maps \( \mathrm{H} \times \mathrm{K} \) into \( \mathrm{K} \); hence (Chapter I, \( §2 \), no. I, Theorem I) it maps \( \mathbf{G} \times \mathbf{K} = \overline{\mathbf{H}} \times \overline{\math... | No |
Proposition 9. Let \( \mathrm{H} \) be a dense subgroup of a topological group \( \mathrm{G} \) . If \( \mathrm{H} \) is generated by every neighbourhood of the identity element in \( \mathrm{H} \), then \( \mathrm{G} \) is generated by every neighbourhood of the identity element in \( \mathrm{G} \) . | Let \( \mathrm{V} \) be any symmetric neighbourhood of \( e \) in \( \mathrm{G} \) . Then \( \mathrm{V} \cap \mathrm{H} \) is a neighbourhood of \( e \) in \( \mathbf{H} \), and hence generates \( \mathbf{H} \) . It follows that \( \mathrm{V} \) generates a subgroup \( {\mathrm{H}}^{\prime } \) which contains \( \mathr... | Yes |
Lemma 2. If a topological group \( \mathrm{G} \) operates continuously on a topological space \( \mathrm{X} \), then the equivalence relation \( \mathrm{R} \) defined by \( \mathrm{G} \) is open. | For the saturation with respect to \( \mathbf{R} \) of an open subset \( \mathbf{U} \) of \( \mathbf{X} \) is the set \( \mathop{\bigcup }\limits_{{s \in \mathrm{G}}}s.\mathrm{U} \), and each \( s.\mathrm{U} \) is open by Lemma 1. | Yes |
Proposition 11. If the equivalence relation \( \mathrm{S} \) on \( \mathrm{X} \) is open and compatible with \( \mathrm{G} \), then \( \mathrm{G} \) operates continuously on \( \mathrm{X}/\mathrm{S} \) . | Since the relation of equality on \( G \) and the relation \( S \) on \( X \) are open, it is enough to show that the mapping \( \left( {s, x}\right) \rightarrow s.\psi \left( x\right) = \psi \left( {s.x}\right) \) of \( \mathrm{G} \times \mathrm{X} \) into \( \mathrm{X}/\mathrm{S} \) is continuous (Chapter I,§ 5, no. ... | Yes |
Proposition 12. The group \( \mathrm{G} \) operates continuously on every homogeneous space \( \mathrm{G}/\mathrm{H} \) . | Since the equivalence relation \( {x}^{-1}y \in \mathrm{H} \) is open (no. 4, Lemma 2) this is a particular case of Proposition II of no. 4. | No |
Proposition 13. Let \( \mathrm{G} \) be a topological group and let \( \mathrm{H} \) be a subgroup of \( \mathrm{G} \). Then the homogeneous space \( \mathrm{G}/\mathrm{H} \) is Hausdorff if and only if \( \mathrm{H} \) is closed in \( \mathrm{G} \). | \( \mathrm{H} \) is an equivalence class for the relation \( {x}^{-1}y \in \mathrm{H} \) and therefore, if \( \mathrm{G}/\mathrm{H} \) is Hausdorff, \( \mathrm{H} \) is closed in \( \mathrm{G} \). Conversely, if \( \mathrm{H} \) is closed, then the graph of this relation is closed in \( \mathrm{G} \times \mathrm{G} \),... | No |
Proposition 14. Let \( \mathbf{G} \) be a topological group and let \( \mathbf{H} \) be a subgroup of \( \mathrm{G} \) . Then the homogeneous space \( \mathrm{G}/\mathrm{H} \) is discrete if and only if \( \mathrm{H} \) is open in \( \mathrm{G} \) . | For the inverse images in \( \mathrm{G} \) of the points of \( \mathrm{G}/\mathrm{H} \) under the canonical mapping are the cosets \( x\mathrm{H}\left( {x \in \mathrm{G}}\right) \) ; and these sets are open in \( \mathrm{G} \) if and only if \( \mathrm{H} \) is open in \( \mathrm{G} \) . | Yes |
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