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A sequence $S$ converges to a point $l$ in a topological space $X$ if and only if $l$ is a point of $X$ and for every open set $U$ containing $l$, there exists an integer $N$ such that for all $n \geq N$, $S(n)$ is in $U$. |
If $f$ converges to $l$ in $X$ along a subsequence of the natural numbers, then $f(n + k)$ converges to $l$ in $X$ along the same subsequence. |
If the sequence $(f(i + k))_{i \in \mathbb{N}}$ converges to $l$ in $X$, then the sequence $(f(i))_{i \in \mathbb{N}}$ converges to $l$ in $X$. |
A function $f$ has a limit $y$ at $x$ if and only if $y$ is in the topological space $Y$ and for every open set $V$ containing $y$, there exists an open set $U$ containing $x$ such that $f(U - \{x\}) \subseteq V$. |
A function $f$ from a topological space $X$ to a topological space $Y$ has a limit at a point $a \in X$ if and only if $f(a) \in Y$ and for every open set $V$ containing $f(a)$, there exists an open set $U$ containing $a$ such that $f(U) \subseteq V$. |
If $F$ is a trivial filter, then $f$ converges to $y$ in $F$. |
If $f$ and $g$ are eventually equal and $f$ converges to $l$ in $X$, then $g$ converges to $l$ in $X$. |
If $g$ is a continuous map from $X$ to $Y$ and $f$ is a limit of $F$ in $X$, then $g \circ f$ is a limit of $F$ in $Y$. |
A function $f$ is topologically continuous at $x$ if and only if $x$ is in the topological space $X$, $f(x)$ is in the topological space $Y$, and the limit of $f$ at $x$ is the same as the limit of $f$ at $x$ in the topological space $X$. |
A map $f$ from a topological space $X$ to a topological space $Y$ is continuous if and only if it is topologically continuous at every point of $X$. |
A map $f$ from a topological space $X$ to a topological space $Y$ is continuous if and only if for every $x \in X$, the limit of $f$ at $x$ exists and is equal to $f(x)$. |
If $f$ is a continuous map from a topological space $X$ to a topological space $Y$, and $a$ is a point in $X$ such that $f(a) = b$, then $f$ has a limit at $a$ equal to $b$. |
The constant function $x \mapsto c$ is continuous. |
If $f$ and $g$ are continuous real-valued functions on a topological space $X$, then so is $f \cdot g$. |
If $f$ is a continuous real-valued function, then $x \mapsto f(x)^n$ is also continuous. |
If $f$ is a continuous real-valued function, then so is $c \cdot f$. |
A map $f$ from a topological space $X$ to the real numbers is continuous if and only if either $f$ is the zero map or $f$ is continuous. |
If $f$ is a continuous real-valued function, then so is $f \cdot c$ for any constant $c$. |
A map $f$ from a topological space $X$ to the real numbers is continuous if and only if $f$ is continuous or $f$ is the zero map. |
If $f$ is a continuous map from $X$ to $\mathbb{R}^n$, then $-f$ is also a continuous map from $X$ to $\mathbb{R}^n$. |
A function $f$ is continuous if and only if the function $-f$ is continuous. |
If $f$ and $g$ are continuous maps from $X$ to $\mathbb{R}^n$, then $f + g$ is a continuous map from $X$ to $\mathbb{R}^n$. |
If $f$ and $g$ are continuous maps from $X$ to $\mathbb{R}^n$, then $f - g$ is a continuous map from $X$ to $\mathbb{R}^n$. |
If $f$ is a continuous map from $X$ to $\mathbb{R}^n$, then the function $x \mapsto \|f(x)\|$ is continuous. |
If $f$ is a continuous real-valued function on a topological space $X$, then $|f|$ is also continuous. |
If $f$ and $g$ are continuous real-valued functions on a topological space $X$, then the function $x \mapsto \max(f(x), g(x))$ is continuous. |
If $f$ and $g$ are continuous real-valued functions on a topological space $X$, then the function $x \mapsto \min(f(x), g(x))$ is continuous. |
If $f$ is a function from a set $X$ to $\mathbb{R}^n$ such that $f$ is continuous for each $i \in \{1, \ldots, n\}$, then the function $x \mapsto \sum_{i=1}^n f_i(x)$ is continuous. |
If $f$ is a function from $X$ to $\mathbb{R}^I$ such that each component function $f_i$ is continuous, then the function $x \mapsto \prod_{i \in I} f_i(x)$ is continuous. |
If $f$ is a continuous map from a topological space $X$ to the real numbers such that $f(x) \neq 0$ for all $x \in X$, then the map $x \mapsto 1/f(x)$ is continuous. |
If $f$ and $g$ are continuous real-valued functions on a topological space $X$ such that $g(x) \neq 0$ for all $x \in X$, then the function $f/g$ is continuous. |
If $S$ is a retract of a contractible space $T$, then $S$ is contractible. |
If $S$ is a retract of a path-connected space $T$, then $S$ is path-connected. |
If $S$ is a retract of a simply connected space $T$, then $S$ is simply connected. |
If $T$ is a retract of $S$ and $f, g: U \to S$ are homotopic in $S$, then $f, g: U \to T$ are homotopic in $T$. |
If $T$ is a retract of $S$ and $f$ is a continuous map from $U$ to $T$, then $f$ is homotopic to a constant map. |
If $r$ is a retraction from $S$ to $T$, then $S \cap r^{-1}(U)$ is open in $S$ if and only if $U$ is open in $T$. |
If $S$ is a locally compact space and $T$ is a retract of $S$, then $T$ is locally compact. |
If $f$ and $g$ are homotopic maps from $S$ to $T$ and $T$ is a retract of $U$, then $f$ and $g$ are homotopic maps from $S$ to $U$. |
If $S$ is a retract of a locally connected space $T$, then $S$ is locally connected. |
If $S$ is a retract of a locally path-connected space $T$, then $S$ is locally path-connected. |
If $S$ is homotopy equivalent to $T$ via a retraction $r$, then $S$ is homotopy equivalent to $T$. |
A space $S$ is a deformation retract of a space $T$ if and only if $T$ is a retract of $S$ and there exists a continuous map $f$ from $S$ to $T$ such that $f$ is homotopic to the identity map on $S$ and $f(S) \subseteq T$. |
If $S$ is a contractible set and $a \in S$, then there exists a retraction $r$ of $S$ onto $\{a\}$. |
If $p$ and $q$ are continuous maps from a compact set $T$ to itself such that $p \circ q = q \circ p = \text{id}_T$, then $q$ is continuous. |
If $S$ is a compact set in a cone $V$ and $S$ is the set of all points of the form $kx$ for $x \in V - \{0\}$ and $k > 0$, then the function $x \mapsto kx$ is continuous on $V - \{0\}$. |
If $f$ and $g$ are bijections from $A$ to $B$ and $a \in A$, and if $f(x) = g(x)$ for all $x \in A$ with $x \neq a$, then $f(a) = g(a)$. |
The image of a set $A$ under the function $f$ that swaps the $i$th and $j$th coordinates of a tuple is equal to the image of $A$ under $f$ if $i$ and $j$ are both in $A$, the image of $A$ under $f$ with $i$ and $j$ swapped if $i$ is in $A$ and $j$ is not, the image of $A$ under $f$ with $i$ and $j$ swapped if $j$ is in... |
The lemma swap_apply(1) is now called swap_apply1. |
The lemma swap_apply(2) is now called swap_apply2. |
If $s$ is a finite set of functions from $\mathbb{N}$ to $\mathbb{N}$ such that for every $x, y \in s$, either $x \leq y$ or $y \leq x$, then there exists a function $a \in s$ such that for every $x \in s$, $a \leq x$, and there exists a function $a \in s$ such that for every $x \in s$, $x \leq a$. |
Let $P$ and $Q$ be predicates on $\mathbb{R}^n$. Suppose that $P$ is closed under the action of $f$ and that $P$ implies that $Q$ is bounded. Then there exists a function $l$ such that $l$ is bounded, $l$ is zero on the boundary of $Q$, $l$ is one on the boundary of $Q$, and $l$ is monotone with respect to $f$. |
If $F$ is a finite set of faces of a simplicial complex $S$, and if $F$ satisfies certain conditions, then the number of simplices in $S$ is odd. |
Suppose we have a finite set of simplices, and we have a function $rl$ that maps each simplex to a subset of $\{0, 1, \ldots, n\}$. Suppose that for each simplex $s$, the image of $rl$ is a subset of $\{0, 1, \ldots, n + 1\}$. Suppose that for each face $f$ of a simplex $s$, if $f$ is a boundary face, then there is exa... |
The set $s$ is equal to the image of the set $\{0, 1, \ldots, n\}$ under the function $enum$. |
If $i$ is less than $n$, then the update function $upd_i$ is less than $n$. |
$s$ is a function from a subset of $\{0, 1, \ldots, n-1\}$ to $\{0, 1, \ldots, p\}$. |
The function upd is injective on the set $\{0, 1, \ldots, n-1\}$. |
The function enum is injective on the set $\{0, 1, \ldots, n\}$. |
The enumeration of the empty set is the empty sequence. |
The base is in the set $s$. |
If $i \leq n$, then $enum(i) \in s$. |
If $a$ is an element of $s$ and $j < n$, and if $a'$ is any element of $s$ such that $a' |
If $i$ and $j$ are less than $n$, then the functions $upd_i$ and $upd_j$ are equal if and only if $i = j$. |
The image of the set $\{0, 1, \ldots, n-1\}$ under the function $upd$ is the set $\{0, 1, \ldots, n-1\}$. |
If $A$ is a subset of $\{0, 1, \ldots, n-1\}$ and $i < n$, then $i \in A$ if and only if $i$ is in the image of $A$ under the function $upd$. |
If $i$ and $j$ are natural numbers less than or equal to $n$, then $enum(i) = enum(j)$ if and only if $i = j$. |
If $A$ is a subset of $\{0, 1, \ldots, n\}$ and $i \leq n$, then $i \in A$ if and only if $i$ is in the image of $A$ under the enumeration function. |
If $i \leq n$ and $j \leq n$, then $enum(i) \leq enum(j)$ if and only if $i \leq j$. |
If $i \leq n$ and $j \leq n$, then $i < j$ if and only if $enum(i) < enum(j)$. |
If $a$ and $b$ are elements of a chain, then either $a \leq b$ or $b \leq a$. |
If $a$ and $b$ are elements of a totally ordered set $s$, and $a_i < b_i$ for all $i$, then $a < b$. |
If $a$ is an element of a set $s$, then $a$ is the least element of $s$ if and only if $a$ is less than or equal to every element of $s$. |
If $a$ is an element of a set $s$, then $a$ is the $n$th element of $s$ if and only if $a$ is greater than or equal to every other element of $s$. |
If $i < n$, then the $i$th element of the enumeration of $n$ is the $i$th element of the enumeration of $i$ plus one. |
If $i \leq n \leq j$, then $enum(i, j) = p$. |
If $a$ is an element of $s$ and $n \leq j$, then $a_j = p$. |
If $a$ is an element of $s$, then $a_j \leq p$. |
If $a$ is an element of $s$, then $a_j \leq b_j + 1$ for all $j$. |
If $a$ is an element of $s$, then $a_j \leq b_j$ for all $j$. |
If $i \leq n$ and $j < n$, then $i \choose j$ is less than or equal to $p$. |
If $a$ is an element of a set $s$ and $i$ is a natural number less than $n$, then $a$ is less than the $i$th element of the enumeration of $s$ if and only if $a$ is less than or equal to the $(i+1)$st element of the enumeration of $s$. |
If $n = 0$, then the $n$-simplex $s$ is the set containing the single point $(\lambda x. p)$. |
If $j < n$, $a \in s$, and $x \in s$, and if $x_j = 0$ for all $x \in s - \{a\}$, then $x \leq a$. |
If $a$ is an element of $s$ and $x$ is an element of $s$ such that $x_j = p$ for all $x \in s - \{a\}$, then $a \leq x$. |
Suppose $i \leq l \leq j$ and $j + d \leq n$. If the sets $\{s_i, \ldots, s_j\}$ and $\{t_{i + d}, \ldots, t_{j + d}\}$ are equal, then $s_l = t_{l + d}$. |
If $s$ and $t$ are simplices such that $s - \{a\} = t - \{b\}$ for some $a \in s$ and $b \in t$, then $s = t$. |
If $s$ and $t$ are simplices such that $s - \{a\} = t - \{b\}$ for some $a \in s$ and $b \in t$, then $s = t$. |
There are finitely many $k$-simplices in an $n$-simplex. |
If $s$ is a $k$-simplex in $\mathbb{R}^n$, then $|s| = k + 1$. |
If $s'$ is a face of a $p$-simplex in $\mathbb{R}^n$ and $s'$ is not the empty set, then $s'$ is the face of a $p$-simplex in $\mathbb{R}^{n+1}$ obtained by removing a vertex. |
If $s$ is a $k$-simplex and $a \in s$, then there is a unique $k$-simplex $s'$ such that $s' - \{b\} = s - \{a\}$ for some $b \in s'$. |
If $s$ is a $k$-simplex, $a$ is a point in $s$, and $j$ is a coordinate such that $x_j = p$ for all $x \in s - \{a\}$, then there is exactly one $k$-simplex $s'$ such that $s' - \{b\} = s - \{a\}$ for some $b \in s'$. |
If $s$ is a $k$-simplex, $a$ is a vertex of $s$, and $s$ has at least two vertices, then there are exactly two $k$-simplices that contain $a$ and have the same vertices as $s$ except for $a$. |
If $S$ is a set of $n+1$ elements, then the number of subsets of $S$ with an odd number of elements is odd. |
The reduced form of a natural number $x$ is a natural number $y$ such that $y \leq x$, $y$ has no leading zeros, and $y$ is the smallest such number. |
If $r \leq n$ and $x_i = 0$ for all $i < r$, and either $r = n$ or $x_r \neq 0$, then $r$ is the reduced form of $x$. |
If $x_j = 0$ and $j < n$, then $j$ is not the reduced labelling of $x$. |
The reduced form of $0$ is $0$. |
If $j < n$ and $x_j \neq 0$, then the reduced index of $x$ is at most $j$. |
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