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If $B$ is a countable set of open sets that forms a basis for the topology, then $B$ is a countable basis for the topology. |
If $X$ is a first-countable topological space, then for every $x \in X$, there exists a countable collection of open sets $\{A_n\}$ such that $x \in A_n$ for all $n$ and for every open set $S$ containing $x$, there exists $n$ such that $A_n \subseteq S$. |
If a topological space is first countable, then there exists a countable basis for the topology at every point. |
If a topological space has a countable base at a point $x$, then it is first-countable at $x$. |
There exists a countable basis for the topology. |
Every second-countable topological space is a countable union of open sets. |
If $\mathcal{F}$ is a collection of open sets, then there exists a countable subcollection $\mathcal{F}'$ such that $\bigcup \mathcal{F}' = \bigcup \mathcal{F}$. |
If $\mathcal{F}$ is a collection of pairwise disjoint open sets, then $\mathcal{F}$ is countable. |
There exists a countable set $B$ such that for all $x, y \in \mathbb{R}$ with $x < y$, there exists $b \in B$ such that $x < b \leq y$. |
For any linearly ordered set $X$ with the topology of a linear order, there exists a countable subset $B$ of $X$ such that for any $x, y \in X$ with $x < y$, there exists $b \in B$ such that $x \leq b < y$. |
Any dense linear order has a countable separating set. |
If every open set containing $x$ contains a point of $S$ other than $x$, then $x$ is a limit point of $S$. |
If $x$ is a limit point of $S$ and $T$ is an open set containing $x$, then there exists a point $y \in S \cap T$ such that $y \neq x$. |
A point $x$ is a limit point of a set $S$ if and only if for every neighborhood $U$ of $x$, there exists a point $y \in U$ such that $y \in S$. |
If $x$ is a limit point of $S$ and $S \subseteq T$, then $x$ is a limit point of $T$. |
A point $x$ is a limit point of the whole space if and only if the singleton $\{x\}$ is not open. |
A point $x$ is a limit point of a set $S$ if and only if $x$ is a limit point of $S$ with $x$ removed. |
Every point in a perfect space is a limit point of the whole space. |
A set $S$ is closed if and only if every limit point of $S$ is in $S$. |
The empty set has no limit points. |
A point $x$ is a limit point of the union of two sets $S$ and $T$ if and only if $x$ is a limit point of $S$ or $x$ is a limit point of $T$. |
A point $x$ is a limit point of a set $S$ if and only if $x$ is a limit point of $S \cup \{a\}$. |
If $s$ is a finite set, then $x$ is not a limit point of $s$. |
If $x$ is a limit point of $s \cup t$ and $s$ is finite, then $x$ is a limit point of $t$. |
A point $l$ is a limit point of a set $S$ if and only if every open neighborhood of $l$ contains infinitely many points of $S$. |
If $l$ is an accumulation point of the range of $f$, then there exists a subsequence of $f$ that converges to $l$. |
If $l$ is a limit point of the range of $f$, then there exists a subsequence of $f$ that converges to $l$. |
If a sequence converges to a limit $l$, then $l$ is the only limit point of the sequence. |
If $T$ is an open set containing $x$ and $T \subseteq S$, then $x$ is an interior point of $S$. |
If $x$ is an interior point of $S$, then there exists an open set $T$ such that $x \in T \subseteq S$. |
The interior of a set is open. |
The interior of a set is a subset of the set. |
If $T$ is an open subset of $S$, then $T$ is contained in the interior of $S$. |
If $S$ is open, then $S$ is its own interior. |
A set $S$ is open if and only if its interior is equal to $S$. |
If $S$ is open and $S \subseteq T$, then $S \subseteq \operatorname{int}(T)$. |
The interior of the empty set is the empty set. |
The interior of the whole space is the whole space. |
The interior of the interior of a set is the interior of the set. |
If $S \subseteq T$, then $\operatorname{int}(S) \subseteq \operatorname{int}(T)$. |
If $T$ is a subset of $S$ and $T$ is open, and if $T'$ is any other open subset of $S$ that is contained in $T$, then $T$ is the interior of $S$. |
The interior of a singleton set is empty. |
The interior of the intersection of two sets is the intersection of their interiors. |
If $x$ is an interior point of $S$, then there is a neighborhood of $x$ that is contained in $S$. |
If $x$ is an interior point of $S$, then $x$ is a limit point of $S$. |
If $S$ is closed and $T$ has empty interior, then the interior of $S \cup T$ is the same as the interior of $S$. |
The interior of the Cartesian product of two sets is the Cartesian product of their interiors. |
If $b < x$, then the interior of the interval $[x, \infty)$ is the interval $(x, \infty)$. |
If $x$ is less than $b$, then the interior of the closed interval $[-\infty, x]$ is the open interval $(-\infty, x)$. |
If the interior of each nonempty set in a pairwise disjoint collection of sets is nonempty, then the collection is countable. |
The interior of a set $S$ is the complement of the closure of the complement of $S$. |
The closure of a set $S$ is the complement of the interior of the complement of $S$. |
The closure of a set is closed. |
The closure of a set $S$ is a superset of $S$. |
The closure of a set is the smallest closed set containing it. |
A set is closed if and only if it is equal to its closure. |
If $S$ is closed, then its closure is $S$. |
The closure of the closure of a set is the closure of the set. |
If $S \subseteq T$, then $\overline{S} \subseteq \overline{T}$. |
If $S$ is a subset of $T$ and $T$ is closed, then the closure of $S$ is a subset of $T$. |
If $S$ is a subset of a closed set $T$, and if $T$ is the smallest closed set containing $S$, then $T$ is the closure of $S$. |
The closure of the empty set is the empty set. |
The closure of the entire space is the entire space. |
The closure of the union of two sets is the union of the closures of the two sets. |
The closure of a set is empty if and only if the set is empty. |
A set $S$ is closed if and only if its closure is a subset of $S$. |
If $S$ is open and $S \cap T = \emptyset$, then $S \cap \overline{T} = \emptyset$. |
If $S$ is open, then $S \cap \overline{T} \subseteq \overline{S \cap T}$. |
The closure of the complement of a set $S$ is the complement of the interior of $S$. |
The interior of the complement of a set $S$ is the complement of the closure of $S$. |
The interior of the difference of two sets is the difference of their interiors. |
The closure of the Cartesian product of two sets is the Cartesian product of their closures. |
If $S$ is an open set and $S \subseteq \overline{T}$, then $\overline{S \cap T} = \overline{S}$. |
The closure of the intersection of a collection of sets is contained in the intersection of the closures of the sets. |
A point $x$ is a limit point of a set $S$ if and only if $x$ is in the closure of $S - \{x\}$. |
If $S$ is connected, then its closure is connected. |
If $A$ is bounded below, then its closure is bounded below. |
The frontier of a set is closed. |
The frontier of a set $S$ is the intersection of the closure of $S$ with the closure of the complement of $S$. |
The frontier of the intersection of two sets is the closure of the intersection of the two sets, intersected with the union of the frontiers of the two sets. |
The frontier of the intersection of two sets is a subset of the union of the frontiers of the two sets. |
If $S$ and $T$ are closed sets, then the frontier of $S \cap T$ is equal to the union of the frontier of $S$ with the frontier of $T$. |
If $S$ is a closed set, then the frontier of $S$ is a subset of $S$. |
The frontier of the empty set is the empty set. |
The frontier of a set $S$ is a subset of $S$ if and only if $S$ is closed. |
The frontier of the complement of a set is the same as the frontier of the set. |
The frontier of the union of two sets is a subset of the union of the frontiers of the two sets. |
The frontier of a set $S$ is disjoint from $S$ if and only if $S$ is open. |
The frontier of the whole space is empty. |
The frontier of a set $s$ is the complement of the union of the interior of $s$ and the interior of the complement of $s$. |
The frontier of the interior of a set is a subset of the frontier of the set. |
The closure of a set $S$ is equal to the union of $S$ and the frontier of $S$. |
A function has a trivial limit at $a$ within $S$ if and only if $a$ is not a limit point of $S$. |
A sequence converges to a point $a$ if and only if $a$ is not a limit point of the sequence. |
The limit at a point $a$ is not trivial. |
A function has a nontrivial limit at $x$ within $S$ if and only if $x$ is in the closure of $S - \{x\}$. |
If $x$ is not in the closure of $s$, then the limit of any function at $x$ within $s$ is trivial. |
If $f$ converges to $l$ at $x$ within $s$, then $f$ converges to $l$ at $x$ within the closure of $s$. |
The filter $at c within A$ is the empty filter if and only if $c$ is not in the closure of $A - \{c\}$. |
If a net converges to a constant, then it eventually satisfies any predicate. |
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