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If $S$ is a bounded set and $f$ is a bounded linear operator, then $f(S)$ is a bounded set. |
If $S$ is a bounded set, then the set of all scalar multiples of elements of $S$ is also bounded. |
If $f$ is a function from $S$ to a real normed vector space, and $f(S)$ is bounded, then $r f(S)$ is bounded for any real number $r$. |
If $S$ is a bounded set, then the set $S + a$ is also bounded. |
If $S$ is a bounded set, then the set of all points obtained by translating $S$ by $a$ is also bounded. |
The set $-X$ is bounded if and only if $X$ is bounded. |
The image of a set $S$ under the function $f$ is bounded if and only if the image of $S$ under the function $-f$ is bounded. |
If $f$ and $g$ are bounded functions, then so is $f + g$. |
If $S$ and $T$ are bounded sets, then the set of sums of elements of $S$ and $T$ is bounded. |
If $f$ and $g$ are bounded functions, then so is $f - g$. |
If $S$ and $T$ are bounded sets, then the set of differences $S - T = \{x - y \mid x \in S, y \in T\}$ is bounded. |
The set of all real vectors is not bounded. |
If the complement of a set $S$ is bounded, then $S$ is unbounded. |
If a sequence of real numbers is summable, then the sequence is bounded. |
If the series $\sum_{n=0}^\infty f(n)$ converges, then the sequence of partial sums $\sum_{n=0}^k f(n)$ is bounded. |
If the power series $\sum_{n=0}^\infty a_n z^n$ converges for $z$, then it converges absolutely for $w$ if $|w| < |z|$. |
The set of all real numbers is not compact. |
The real numbers are not a compact space. |
If $S$ is a closed set, then there exists a sequence of compact sets $F_n$ such that $F_n \subseteq S$, $F_n \subseteq F_{n+1}$, and $\bigcup_n F_n = S$. |
If $\mathcal{F}$ is a chain of closed sets with a bounded element, then $\bigcap \mathcal{F} \neq \emptyset$. |
If $\mathcal{F}$ is a chain of compact sets, then $\bigcap \mathcal{F}$ is nonempty. |
If $F_n$ is a sequence of nonempty compact sets such that $F_n \subseteq F_m$ whenever $n \leq m$, then $\bigcap_{n=1}^\infty F_n$ is nonempty. |
If $S$ is a closed set and $\mathcal{G}$ is a countable collection of open sets whose closures cover $S$, then $S$ is contained in the closure of the intersection of the sets in $\mathcal{G}$. |
If $g$ is uniformly continuous on $s$, then $f \circ g$ is uniformly continuous on $s$. |
If $f$ and $g$ are uniformly continuous functions on a set $S$, then the function $x \mapsto \|f(x) - g(x)\|$ is uniformly continuous on $S$. |
If $f$ is uniformly continuous on $S$, then the function $x \mapsto f(x)c$ is uniformly continuous on $S$. |
If $f$ is uniformly continuous on $S$, then $c \cdot f$ is uniformly continuous on $S$. |
If $f$ is uniformly continuous on $s$, then $\|f\|$ is uniformly continuous on $s$. |
If $f$ is uniformly continuous on $S$, then $cf$ is uniformly continuous on $S$. |
The distance between two points is the same as the distance between their negatives. |
If $f$ is uniformly continuous on $s$, then so is $-f$. |
If $f$ and $g$ are uniformly continuous on a set $S$, then $f + g$ is uniformly continuous on $S$. |
If $f$ and $g$ are uniformly continuous on a set $S$, then $f - g$ is uniformly continuous on $S$. |
If $s$ is an open set and $c \neq 0$, then the set $c \cdot s$ is open. |
For any set $A$, the image of $A$ under the function $x \mapsto -x$ is equal to the preimage of $A$ under the function $x \mapsto -x$. |
If $S$ is an open set, then so is the set of all negations of elements of $S$. |
If $S$ is an open set, then the set $S + a$ is open. |
If $S$ is an open set, then the set $S - a$ is open. |
If $S$ is an open set, then the set of all points $a - x$ for $x \in S$ is also open. |
If $S$ is an open set and $c \neq 0$, then the set $a + cS = \{a + cx : x \in S\}$ is open. |
The interior of a translation of a set is the translation of the interior of the set. |
The interior of a translation of a set is the translation of the interior of the set. |
If $s$ is a compact set, then the set of all scalar multiples of elements of $s$ is compact. |
If $s$ is a compact set, then the set of all negations of elements of $s$ is compact. |
If $s$ and $t$ are compact sets, then the set of all sums $x + y$ where $x \in s$ and $y \in t$ is compact. |
If $s$ and $t$ are compact sets, then the set of all differences $x - y$ where $x \in s$ and $y \in t$ is compact. |
If $s$ is a compact set, then $s + a$ is also compact for any $a \in \mathbb{R}^n$. |
If $s$ is a compact set, then the set $\{x - a \mid x \in s\}$ is compact. |
If $s$ is a compact set, then the image of $s$ under the affine map $x \mapsto a + cx$ is also compact. |
If $S$ is a closed set, then the set of all scalar multiples of elements of $S$ is also closed. |
If $S$ is a closed set, then the set of all negations of elements of $S$ is also closed. |
If $S$ is compact and $T$ is closed, then the set $\{x + y \mid x \in S, y \in T\}$ is closed. |
If $S$ is closed and $T$ is compact, then the set $\{x + y : x \in S, y \in T\}$ is closed. |
If $S$ is compact and $T$ is closed, then the set of differences $S - T = \{x - y \mid x \in S, y \in T\}$ is closed. |
If $S$ is closed and $T$ is compact, then the set of all differences $x - y$ where $x \in S$ and $y \in T$ is closed. |
The translation of a closed set is closed. |
The translation of a closed set by a vector is closed. |
The closure of a translation of a set is the translation of the closure of the set. |
The closure of the set of all points $x - a$ for $x \in S$ is the set of all points $x - a$ for $x \in \overline{S}$. |
The frontier of a translation of a set is the translation of the frontier of the set. |
The frontier of a translation of a set is the translation of the frontier of the set. |
The sphere of radius $r$ centered at $a$ is the image of the sphere of radius $r$ centered at $0$ under the translation $x \mapsto x + a$. |
The sphere of radius $r$ centered at $c - a$ is the image of the sphere of radius $r$ centered at $c$ under the translation $x \mapsto x - a$. |
The ball of radius $r$ centered at $a$ is the image of the ball of radius $r$ centered at $0$ under the translation $x \mapsto x + a$. |
The ball of radius $r$ centered at $c - a$ is the image of the ball of radius $r$ centered at $c$ under the translation $x \mapsto x - a$. |
The ball of radius $r$ centered at $a$ is the image of the ball of radius $r$ centered at $0$ under the translation $x \mapsto x + a$. |
The ball of radius $r$ centered at $c - a$ is the image of the ball of radius $r$ centered at $c$ under the translation $x \mapsto x - a$. |
If $S$ is a set in a real vector space, then $S$ is homeomorphic to the set $cS$ obtained by scaling $S$ by a nonzero constant $c$. |
Any set $S$ is homeomorphic to its translation by any vector $a$. |
If $S$ is a set in a real vector space and $c \neq 0$, then $S$ is homeomorphic to the set $a + cS = \{a + cx : x \in S\}$. |
For any two positive real numbers $d$ and $e$, the open balls of radius $d$ and $e$ are homeomorphic. The same is true for closed balls. |
If $d$ and $e$ are positive real numbers, then the spheres of radius $d$ and $e$ centered at $a$ and $b$ are homeomorphic. |
The unit ball in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. |
For any $r > 0$, the open ball of radius $r$ centered at $a$ is homeomorphic to $\mathbb{R}^n$. |
If $f$ is a function from a set $S$ to a finite set, then $f$ is discrete. |
If $f$ is a bounded linear map from a subspace $s$ of a normed vector space to itself such that $\|f(x)\| \geq e \|x\|$ for all $x \in s$ and $e > 0$, and if $(x_n)$ is a Cauchy sequence in $s$ such that $f(x_n)$ is a Cauchy sequence, then $(x_n)$ is a Cauchy sequence. |
If $f$ is a bounded linear map from a complete normed vector space $s$ to a normed vector space $t$ such that $\|f(x)\| \geq e \|x\|$ for all $x \in s$, then $f(s)$ is complete. |
If $s$ is a compact set and $c$ is a component of $s$, then $c$ is compact. |
If $f$ is a continuous function from a set $S$ to a normed vector space $V$ such that for every $x \in S$, there exists an $\epsilon > 0$ such that for all $y \in S$ with $f(y) \neq f(x)$, we have $\epsilon \leq \|f(y) - f(x)\|$, then the image of $f$ is a discrete set. |
If $S$ is a connected set, then any continuous function $f$ defined on $S$ with a disconnected range, a discrete range, or a finite range is constant. |
If $f$ is a continuous function from a connected set $S$ to a normed vector space, and for every $x \in S$, there exists an $\epsilon > 0$ such that for all $y \in S$, if $f(y) \neq f(x)$, then $\epsilon \leq \|f(y) - f(x)\|$, then $f$ is constant on $S$. |
If $f$ is a continuous function from a connected set $S$ to a normed vector space $V$ and the range of $f$ is finite, then $f$ is constant. |
If $m > 0$, then $m x + c \leq y$ if and only if $x \leq \frac{1}{m} y - \frac{c}{m}$. |
If $m > 0$, then $y \leq mx + c$ if and only if $\frac{y}{m} - \frac{c}{m} \leq x$. |
If $m > 0$, then $m x + c < y$ if and only if $x < \frac{1}{m} y - \frac{c}{m}$. |
If $m > 0$, then $y < mx + c$ if and only if $\frac{y}{m} - \frac{c}{m} < x$. |
If $m \neq 0$, then $m x + c = y$ if and only if $x = \frac{1}{m} y - \frac{c}{m}$. |
If $m \neq 0$, then $y = mx + c$ if and only if $\frac{1}{m}y - \frac{c}{m} = x$. |
A set $B$ is a topological basis if and only if every open set is a union of elements of $B$. |
A set $B$ is a topological basis if and only if for every open set $O$, every point $x \in O$ is contained in some element of $B$ that is contained in $O$. |
If $B$ is a collection of open sets such that every open set is a union of elements of $B$ and every element of $B$ is contained in some open set, then $B$ is a topological basis. |
If $B$ is a topological basis for a topology $\tau$, then for any open set $O'$ and any point $x \in O'$, there exists a basis element $B' \in B$ such that $x \in B'$ and $B' \subseteq O'$. |
If $B$ is a topological basis for a topology $\tau$, and $X \in B$, then $X$ is open. |
If $B$ is a topological basis, then the open sets are the sets generated by $B$. |
If $B$ is a topological basis and $f$ is a function that chooses an element from each nonempty set in $B$, then for every nonempty open set $X$, there exists a set $B' \in B$ such that $f(B') \in X$. |
If $A$ and $B$ are topological bases for topologies $\tau_A$ and $\tau_B$, respectively, then $\{a \times b \mid a \in A, b \in B\}$ is a topological basis for the product topology $\tau_A \times \tau_B$. |
If $X$ is a topological space with a countable basis $B$, then there exists a countable subset $B'$ of $B$ such that $X = \bigcup B'$. |
If $X$ is an open set in a topological space with a countable basis $B$, then there exists a subset $B' \subseteq B$ such that $X = \bigcup B'$. |
There exists a countable set $D$ such that for any set $X$ satisfying $p(X)$, if $X$ is nonempty, then there exists $d \in D$ such that $d \in X$. |
If $p$ is a property of non-empty subsets of a set $X$, then there exists a countable set $D$ such that for any non-empty subset $X$ of $X$ satisfying $p$, there exists $d \in D$ such that $d \in X$. |
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