UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
The empty set is totally bounded.
If $S$ is totally bounded, then any subset $T$ of $S$ is also totally bounded.
If $M$ is a finite set of totally bounded subsets of a metric space, then $\bigcup M$ is totally bounded.
A sequence $X$ is Cauchy if and only if for every entourage $P$, there exists an $N$ such that for all $n, m \geq N$, $(X_n, X_m) \in P$.
If $F$ is a filter on a metric space $X$ such that $F \leq \mathcal{N}(x)$ for some $x \in X$, then $F$ is a Cauchy filter.
If a sequence converges, then it is Cauchy.
If $X$ is a Cauchy sequence and $f$ is a strictly monotone function, then $X \circ f$ is a Cauchy sequence.
If a sequence converges, then it is Cauchy.
If $f$ is uniformly continuous on $s$, then for any $\epsilon > 0$, there exists $\delta > 0$ such that for all $x, y \in s$, if $\|x - y\| < \delta$, then $\|f(x) - f(y)\| < \epsilon$.
The constant function $f(x) = c$ is uniformly continuous on any set $S$.
The identity function is uniformly continuous on any set.
If $f$ is uniformly continuous on $g(S)$ and $g$ is uniformly continuous on $S$, then $f \circ g$ is uniformly continuous on $S$.
If $f$ is uniformly continuous on $S$, then $f$ is continuous on $S$.
If $S$ is an open set and $x \in S$, then there exist open sets $A$ and $B$ such that $x \in A \times B$ and $A \times B \subseteq S$.
If for every $x \in S$, there exist open sets $A$ and $B$ such that $x \in A \times B$ and $A \times B \subseteq S$, then $S$ is open.
If $S$ and $T$ are open sets, then $S \times T$ is open.
The preimage of a set $S$ under the first projection is $S \times \mathbb{R}$.
The preimage of a set $S$ under the second projection is the set of all pairs whose second component is in $S$.
If $S$ is an open set, then the preimage of $S$ under the first projection is open.
If $S$ is an open set, then the inverse image of $S$ under the second projection is open.
If $S$ is a closed set, then the preimage of $S$ under the first projection is closed.
If $S$ is a closed set, then the preimage of $S$ under the second projection is closed.
The product of two closed sets is closed.
If $A \times B \subseteq S$ and $y \in B$, then $A \subseteq \text{fst}(S)$.
If $A \times B$ is a subset of $S$, and $x \in A$, then $B$ is a subset of the image of $S$ under the second projection.
If $S$ is an open set in $\mathbb{R}^2$, then the projection of $S$ onto the first coordinate is open in $\mathbb{R}$.
If $S$ is an open set in $\mathbb{R}^2$, then the projection of $S$ onto the second coordinate is open.
The product topology on $\mathbb{R}^2$ is the product of the topologies on $\mathbb{R}$.
If $f$ converges to $a$, then the first component of $f$ converges to the first component of $a$.
If $f$ converges to $a$, then the second component of $f$ converges to the second component of $a$.
If $f$ and $g$ converge to $a$ and $b$, respectively, then $(f, g)$ converges to $(a, b)$.
If $f$ is a continuous function from a topological space $F$ to a topological space $G$, then the function $x \mapsto f(x)_1$ is continuous.
If $f$ is a continuous function from a topological space $X$ to a topological space $Y$, then the function $g$ defined by $g(x) = f(x)_2$ is also continuous.
If $f$ and $g$ are continuous functions, then so is the function $(x, y) \mapsto (f(x), g(y))$.
If $f$ is continuous on $s$, then the function $x \mapsto f(x)_1$ is continuous on $s$.
If $f$ is a continuous function from $S$ to $\mathbb{R}^2$, then the function $x \mapsto f(x)_2$ is continuous.
If $f$ and $g$ are continuous functions on a set $S$, then the function $(x, y) \mapsto (f(x), g(y))$ is continuous on $S$.
The function that swaps the components of a pair is continuous.
If $d$ is a continuous function on $A \times B$, then $d$ is also continuous on $B \times A$.
If $f$ is continuous at $a$, then the function $x \mapsto \text{fst}(f(x))$ is continuous at $a$.
If $f$ is continuous at $a$, then the function $x \mapsto \text{snd}(f(x))$ is continuous at $a$.
If $f$ and $g$ are continuous at $a$, then the function $(x \mapsto (f(x), g(x)))$ is continuous at $a$.
If $f$ is continuous on the product space $A \times B$, and $g$ and $h$ are continuous on $C$, and $g(c) \in A$ and $h(c) \in B(g(c))$ for all $c \in C$, then the function $c \mapsto f(g(c), h(c))$ is continuous on $C$.
If $S$ and $T$ are connected, then $S \times T$ is connected.
A product of two connected sets is connected if and only if one of the sets is empty or both sets are connected.
The function $f(x, y) = (y, x)$ is continuous.
The diagonal complement is open.
The diagonal of a topological space is closed.
The set of all pairs $(x,y)$ of real numbers such that $x > y$ is open.
The set of pairs $(x,y)$ such that $x \leq y$ is closed.
The set of pairs $(x,y)$ such that $x < y$ is open.
The set of all pairs $(x,y)$ of real numbers such that $x \geq y$ is closed.
The Euclidean distance between two points $x$ and $y$ is equal to the $L^2$ distance between the coordinates of $x$ and $y$.
The norm of the $i$th component of a vector is less than or equal to the norm of the vector.
The vectors in the standard basis of $\mathbb{R}^n$ are pairwise orthogonal.
If $B$ is a finite set of vectors that are pairwise orthogonal and independent, then there exists a constant $m > 0$ such that for any vector $x$ in the span of $B$, the representation of $x$ in terms of $B$ has a coefficient of $b$ that is bounded by $m \\|x\\|$.
If $B$ is a finite set of independent vectors that are pairwise orthogonal, then the function $f(x) = \langle x, b \rangle$ is continuous on the span of $B$.
The closed ball of radius $r$ centered at $a$ is contained in the closed ball of radius $r'$ centered at $a'$ if and only if $r < 0$ or $r + d(a, a') \leq r'$.
A closed ball is contained in an open ball if and only if the center of the closed ball is closer to the center of the open ball than the radius of the open ball, or the radius of the closed ball is negative.
The ball of radius $r$ centered at $a$ is contained in the ball of radius $r'$ centered at $a'$ if and only if $r \leq 0$ or $d(a, a') + r \leq r'$.
The ball of radius $r$ centered at $a$ is contained in the ball of radius $r'$ centered at $a'$ if and only if $r \leq 0$ or $d(a, a') + r \leq r'$.
Two balls are equal if and only if they are both empty or they have the same center and radius.
Two closed balls in Euclidean space are equal if and only if they have the same center and radius.
The ball of radius $d$ centered at $x$ is equal to the closed ball of radius $e$ centered at $y$ if and only if $d \leq 0$ and $e < 0$.
The closed ball of radius $d$ centered at $x$ is equal to the open ball of radius $e$ centered at $y$ if and only if $d < 0$ and $e \leq 0$.
If $S$ is an open set and $X$ is a finite set, then there exists an open ball around $p$ that is contained in $S$ and does not contain any points of $X$ other than $p$.
If $S$ is an open set and $X$ is a finite set, then there exists an open ball around $p$ that is contained in $S$ and does not contain any points of $X$ other than $p$.
The dimension of a closed ball in $\mathbb{R}^n$ is $n$.
The vector $1$ is not the zero vector.
The multiplicative identity is not the additive identity.
$0 \neq 1$.
The box $[a, b]$ is the set of all points $x$ such that $a_i < x_i < b_i$ for all $i$.
The Cartesian product of two closed intervals is a closed rectangle.
A point $(x, y)$ is in the closed box $[a, b] \times [c, d]$ if and only if $x \in [a, b]$ and $y \in [c, d]$.
The Cartesian product of two closed intervals in $\mathbb{R}$ is equal to the Cartesian product of two closed intervals in $\mathbb{C}$.
The Cartesian product of two intervals is empty if and only if at least one of the intervals is empty.
The image of a box in $\mathbb{R}^2$ under the swap map is a box in $\mathbb{R}^2$.
For real numbers $a$ and $b$, $x \in [a,b]$ if and only if $a \leq x \leq b$.
For any real numbers $a$ and $b$, the open interval $(a,b)$ is equal to the set $\{x \in \mathbb{R} \mid a < x < b\}$, and the closed interval $[a,b]$ is equal to the set $\{x \in \mathbb{R} \mid a \leq x \leq b\}$.
The intersection of two boxes is a box.
For any point $x$ in $\mathbb{R}^n$ and any $\epsilon > 0$, there exists a box $B$ centered at $x$ such that $B$ is contained in the ball of radius $\epsilon$ centered at $x$ and the coordinates of the center and the corners of $B$ are all rational numbers.
If $M$ is an open set in $\mathbb{R}^n$, then $M$ is the union of a countable collection of open boxes.
Any open set in $\mathbb{R}^n$ can be written as a countable union of open boxes.
For every $x \in \mathbb{R}^n$ and every $\epsilon > 0$, there exists a box $B$ with rational coordinates such that $x \in B$ and $B \subseteq B(x, \epsilon)$.
If $M$ is an open set in $\mathbb{R}^n$, then $M$ can be written as a union of closed boxes.
If $S$ is an open set, then there exists a countable collection of open boxes whose union is $S$.
The box $[a,b]$ is empty if and only if there exists some $i \in \{1,\ldots,n\}$ such that $b_i \leq a_i$. The closed box $[a,b]$ is empty if and only if there exists some $i \in \{1,\ldots,n\}$ such that $b_i < a_i$.
A closed interval $[a,b]$ is non-empty if and only if $a \leq b$ in every coordinate. An open interval $(a,b)$ is non-empty if and only if $a < b$ in every coordinate.
The closed box and open box with the same endpoints are the singleton set and the empty set, respectively.
If $c$ and $d$ are vectors in $\mathbb{R}^n$ such that $a \leq c$ and $d \leq b$, then the closed box $[c,d]$ is contained in the closed box $[a,b]$.
The box $[a,b]$ is contained in the closed box $[a,b]$.
The following four statements are equivalent: 1. $[c,d] \subseteq [a,b]$ 2. $[c,d] \subseteq (a,b)$ 3. $(c,d) \subseteq [a,b]$ 4. $(c,d) \subseteq (a,b)$
Two closed intervals are equal if and only if they are both empty or they have the same endpoints.
The closed box $[a, b]$ is equal to the open box $(c, d)$ if and only if both boxes are empty.
The box $[a,b]$ is equal to the closed box $[c,d]$ if and only if both boxes are empty.
Two boxes are equal if and only if they are both empty or they have the same endpoints.
If $a$ and $b$ are complex numbers, then the closed box $[a,b]$ is contained in the closed box $[c,d]$ if and only if $a \leq b$ and $a \geq c$ and $b \leq d$. The open box $(a,b)$ is contained in the closed box $[c,d]$ if and only if $a < b$ and $a > c$ and $b < d$. The closed box $[a,b]$ is contained in the open box $(c,d)$ if and only if $a \leq b$ and $a \geq c$ and $b \leq d$. The open box $(a,b)$ is contained in the open box $(c,d)$ if and only if $a < b$ and $a > c$ and $b < d$.
A complex number $x$ is in the closed box $[a,b]$ if and only if its real and imaginary parts are in the corresponding closed intervals.
The box product of two complex numbers is the image of the box product of their real and imaginary parts under the map $(x,y) \mapsto x + iy$.
A complex number $x$ is in the box $[a,b]$ if and only if its real and imaginary parts are in the intervals $[\Re(a),\Re(b)]$ and $[\Im(a),\Im(b)]$, respectively.

The empty set is totally bounded.

If $S$ is totally bounded, then any subset $T$ of $S$ is also totally bounded.

If $M$ is a finite set of totally bounded subsets of a metric space, then $\bigcup M$ is totally bounded.

A sequence $X$ is Cauchy if and only if for every entourage $P$, there exists an $N$ such that for all $n, m \geq N$, $(X_n, X_m) \in P$.

If $F$ is a filter on a metric space $X$ such that $F \leq \mathcal{N}(x)$ for some $x \in X$, then $F$ is a Cauchy filter.

If a sequence converges, then it is Cauchy.

If $X$ is a Cauchy sequence and $f$ is a strictly monotone function, then $X \circ f$ is a Cauchy sequence.

If a sequence converges, then it is Cauchy.

If $f$ is uniformly continuous on $s$, then for any $\epsilon > 0$, there exists $\delta > 0$ such that for all $x, y \in s$, if $|x - y| < \delta$, then $|f(x) - f(y)| < \epsilon$.

The constant function $f(x) = c$ is uniformly continuous on any set $S$.

The identity function is uniformly continuous on any set.

If $f$ is uniformly continuous on $g(S)$ and $g$ is uniformly continuous on $S$, then $f \circ g$ is uniformly continuous on $S$.

If $f$ is uniformly continuous on $S$, then $f$ is continuous on $S$.

If $S$ is an open set and $x \in S$, then there exist open sets $A$ and $B$ such that $x \in A \times B$ and $A \times B \subseteq S$.

If for every $x \in S$, there exist open sets $A$ and $B$ such that $x \in A \times B$ and $A \times B \subseteq S$, then $S$ is open.

If $S$ and $T$ are open sets, then $S \times T$ is open.

The preimage of a set $S$ under the first projection is $S \times \mathbb{R}$.

The preimage of a set $S$ under the second projection is the set of all pairs whose second component is in $S$.

If $S$ is an open set, then the preimage of $S$ under the first projection is open.

If $S$ is an open set, then the inverse image of $S$ under the second projection is open.

If $S$ is a closed set, then the preimage of $S$ under the first projection is closed.

If $S$ is a closed set, then the preimage of $S$ under the second projection is closed.

The product of two closed sets is closed.

If $A \times B \subseteq S$ and $y \in B$, then $A \subseteq \text{fst}(S)$.

If $A \times B$ is a subset of $S$, and $x \in A$, then $B$ is a subset of the image of $S$ under the second projection.

If $S$ is an open set in $\mathbb{R}^2$, then the projection of $S$ onto the first coordinate is open in $\mathbb{R}$.

If $S$ is an open set in $\mathbb{R}^2$, then the projection of $S$ onto the second coordinate is open.

The product topology on $\mathbb{R}^2$ is the product of the topologies on $\mathbb{R}$.

If $f$ converges to $a$, then the first component of $f$ converges to the first component of $a$.

If $f$ converges to $a$, then the second component of $f$ converges to the second component of $a$.

If $f$ and $g$ converge to $a$ and $b$, respectively, then $(f, g)$ converges to $(a, b)$.

If $f$ is a continuous function from a topological space $F$ to a topological space $G$, then the function $x \mapsto f(x)_1$ is continuous.

If $f$ is a continuous function from a topological space $X$ to a topological space $Y$, then the function $g$ defined by $g(x) = f(x)_2$ is also continuous.

If $f$ and $g$ are continuous functions, then so is the function $(x, y) \mapsto (f(x), g(y))$.

If $f$ is continuous on $s$, then the function $x \mapsto f(x)_1$ is continuous on $s$.

If $f$ is a continuous function from $S$ to $\mathbb{R}^2$, then the function $x \mapsto f(x)_2$ is continuous.

If $f$ and $g$ are continuous functions on a set $S$, then the function $(x, y) \mapsto (f(x), g(y))$ is continuous on $S$.

The function that swaps the components of a pair is continuous.

If $d$ is a continuous function on $A \times B$, then $d$ is also continuous on $B \times A$.

If $f$ is continuous at $a$, then the function $x \mapsto \text{fst}(f(x))$ is continuous at $a$.

If $f$ is continuous at $a$, then the function $x \mapsto \text{snd}(f(x))$ is continuous at $a$.

If $f$ and $g$ are continuous at $a$, then the function $(x \mapsto (f(x), g(x)))$ is continuous at $a$.

If $f$ is continuous on the product space $A \times B$, and $g$ and $h$ are continuous on $C$, and $g(c) \in A$ and $h(c) \in B(g(c))$ for all $c \in C$, then the function $c \mapsto f(g(c), h(c))$ is continuous on $C$.

If $S$ and $T$ are connected, then $S \times T$ is connected.

A product of two connected sets is connected if and only if one of the sets is empty or both sets are connected.

The function $f(x, y) = (y, x)$ is continuous.

The diagonal complement is open.

The diagonal of a topological space is closed.

The set of all pairs $(x,y)$ of real numbers such that $x > y$ is open.

The set of pairs $(x,y)$ such that $x \leq y$ is closed.

The set of pairs $(x,y)$ such that $x < y$ is open.

The set of all pairs $(x,y)$ of real numbers such that $x \geq y$ is closed.

The Euclidean distance between two points $x$ and $y$ is equal to the $L^2$ distance between the coordinates of $x$ and $y$.

The norm of the $i$th component of a vector is less than or equal to the norm of the vector.

The vectors in the standard basis of $\mathbb{R}^n$ are pairwise orthogonal.

If $B$ is a finite set of vectors that are pairwise orthogonal and independent, then there exists a constant $m > 0$ such that for any vector $x$ in the span of $B$, the representation of $x$ in terms of $B$ has a coefficient of $b$ that is bounded by $m \|x\|$.

If $B$ is a finite set of independent vectors that are pairwise orthogonal, then the function $f(x) = \langle x, b \rangle$ is continuous on the span of $B$.

The closed ball of radius $r$ centered at $a$ is contained in the closed ball of radius $r'$ centered at $a'$ if and only if $r < 0$ or $r + d(a, a') \leq r'$.

A closed ball is contained in an open ball if and only if the center of the closed ball is closer to the center of the open ball than the radius of the open ball, or the radius of the closed ball is negative.

The ball of radius $r$ centered at $a$ is contained in the ball of radius $r'$ centered at $a'$ if and only if $r \leq 0$ or $d(a, a') + r \leq r'$.

Two balls are equal if and only if they are both empty or they have the same center and radius.

Two closed balls in Euclidean space are equal if and only if they have the same center and radius.

The ball of radius $d$ centered at $x$ is equal to the closed ball of radius $e$ centered at $y$ if and only if $d \leq 0$ and $e < 0$.

The closed ball of radius $d$ centered at $x$ is equal to the open ball of radius $e$ centered at $y$ if and only if $d < 0$ and $e \leq 0$.

If $S$ is an open set and $X$ is a finite set, then there exists an open ball around $p$ that is contained in $S$ and does not contain any points of $X$ other than $p$.

The dimension of a closed ball in $\mathbb{R}^n$ is $n$.

The vector $1$ is not the zero vector.

The multiplicative identity is not the additive identity.

$0 \neq 1$.

The box $[a, b]$ is the set of all points $x$ such that $a_i < x_i < b_i$ for all $i$.

The Cartesian product of two closed intervals is a closed rectangle.

A point $(x, y)$ is in the closed box $[a, b] \times [c, d]$ if and only if $x \in [a, b]$ and $y \in [c, d]$.

The Cartesian product of two closed intervals in $\mathbb{R}$ is equal to the Cartesian product of two closed intervals in $\mathbb{C}$.

The Cartesian product of two intervals is empty if and only if at least one of the intervals is empty.

The image of a box in $\mathbb{R}^2$ under the swap map is a box in $\mathbb{R}^2$.

For real numbers $a$ and $b$, $x \in [a,b]$ if and only if $a \leq x \leq b$.

For any real numbers $a$ and $b$, the open interval $(a,b)$ is equal to the set $\{x \in \mathbb{R} \mid a < x < b\}$, and the closed interval $[a,b]$ is equal to the set $\{x \in \mathbb{R} \mid a \leq x \leq b\}$.

The intersection of two boxes is a box.

For any point $x$ in $\mathbb{R}^n$ and any $\epsilon > 0$, there exists a box $B$ centered at $x$ such that $B$ is contained in the ball of radius $\epsilon$ centered at $x$ and the coordinates of the center and the corners of $B$ are all rational numbers.

If $M$ is an open set in $\mathbb{R}^n$, then $M$ is the union of a countable collection of open boxes.

Any open set in $\mathbb{R}^n$ can be written as a countable union of open boxes.

For every $x \in \mathbb{R}^n$ and every $\epsilon > 0$, there exists a box $B$ with rational coordinates such that $x \in B$ and $B \subseteq B(x, \epsilon)$.

If $M$ is an open set in $\mathbb{R}^n$, then $M$ can be written as a union of closed boxes.

If $S$ is an open set, then there exists a countable collection of open boxes whose union is $S$.

The box $[a,b]$ is empty if and only if there exists some $i \in \{1,\ldots,n\}$ such that $b_i \leq a_i$. The closed box $[a,b]$ is empty if and only if there exists some $i \in \{1,\ldots,n\}$ such that $b_i < a_i$.

A closed interval $[a,b]$ is non-empty if and only if $a \leq b$ in every coordinate. An open interval $(a,b)$ is non-empty if and only if $a < b$ in every coordinate.

The closed box and open box with the same endpoints are the singleton set and the empty set, respectively.

If $c$ and $d$ are vectors in $\mathbb{R}^n$ such that $a \leq c$ and $d \leq b$, then the closed box $[c,d]$ is contained in the closed box $[a,b]$.

The box $[a,b]$ is contained in the closed box $[a,b]$.

The following four statements are equivalent: 1. $[c,d] \subseteq [a,b]$ 2. $[c,d] \subseteq (a,b)$ 3. $(c,d) \subseteq [a,b]$ 4. $(c,d) \subseteq (a,b)$

Two closed intervals are equal if and only if they are both empty or they have the same endpoints.

The closed box $[a, b]$ is equal to the open box $(c, d)$ if and only if both boxes are empty.

The box $[a,b]$ is equal to the closed box $[c,d]$ if and only if both boxes are empty.

Two boxes are equal if and only if they are both empty or they have the same endpoints.

If $a$ and $b$ are complex numbers, then the closed box $[a,b]$ is contained in the closed box $[c,d]$ if and only if $a \leq b$ and $a \geq c$ and $b \leq d$. The open box $(a,b)$ is contained in the closed box $[c,d]$ if and only if $a < b$ and $a > c$ and $b < d$. The closed box $[a,b]$ is contained in the open box $(c,d)$ if and only if $a \leq b$ and $a \geq c$ and $b \leq d$. The open box $(a,b)$ is contained in the open box $(c,d)$ if and only if $a < b$ and $a > c$ and $b < d$.

A complex number $x$ is in the closed box $[a,b]$ if and only if its real and imaginary parts are in the corresponding closed intervals.

The box product of two complex numbers is the image of the box product of their real and imaginary parts under the map $(x,y) \mapsto x + iy$.

A complex number $x$ is in the box $[a,b]$ if and only if its real and imaginary parts are in the intervals $[\Re(a),\Re(b)]$ and $[\Im(a),\Im(b)]$, respectively.