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

Statement: stringlengths 7 24.3k
If $f$ is continuous at $a$ from the right, continuous at $b$ from the left, and continuous on $(a,b)$, then $f$ is continuous on $[a,b]$.
If $f$ is continuous on the closed interval $[a, b]$, then $f$ is continuous at $a$ from the right and at $b$ from the left.
A function from a discrete space to any space is continuous.
The constant function $f(x) = \bot$ is continuous.
If a net has a trivial limit, then any function on the net is continuous.
A function $f$ is continuous at $x$ within $S$ if and only if the limit of $f$ at $x$ within $S$ exists and is equal to $f(x)$.
A function $f$ is continuous at $x$ within $S$ if and only if for every open set $B$ containing $f(x)$, there exists an open set $A$ containing $x$ such that $f(y) \in B$ for all $y \in S \cap A$.
If $f$ is continuous at $x$ within $s$ and $g$ is continuous at $f(x)$ within $f(s)$, then $g \circ f$ is continuous at $x$ within $s$.
If $f$ is continuous at $x$ on $S$ and $g$ is continuous at $f(x)$ on $f(S)$, then $g \circ f$ is continuous at $x$ on $S$.
A function $f$ is continuous at $x$ if and only if $f$ converges to $f(x)$ as $x$ approaches $x$.
The identity function is continuous.
The identity function is continuous.
The constant function $x \mapsto c$ is continuous.
A function $f$ is continuous on a set $S$ if and only if it is continuous at every point of $S$.
The function $f$ is continuous at every point $x$ in the discrete topology.
A function $f$ is continuous at $a$ if and only if $f$ has a limit at $a$.
If $f$ is continuous at $x$, then $f$ is defined at $x$.
If $f$ and $g$ are two functions that agree on a neighborhood of $x$, then $f$ is continuous at $x$ if and only if $g$ is continuous at $x$.
If $f$ is continuous at $x$, then $f$ is continuous at $x$ within $s$.
If $f$ is continuous on an open set $S$, then $f$ is continuous at every point of $S$.
If $f$ is continuous at $a$ within $A$, then $f$ is continuous at $a$.
If $f$ is continuous at every point of $s$, then $f$ is continuous on $s$.
If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$.
If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$.
If $g$ is continuous at $l$ and $f$ tends to $l$, then $g \circ f$ tends to $g(l)$.
If $f$ is continuous on $s$, $g$ tends to $l$ in $F$, $l$ is in $s$, and $g$ is eventually in $s$, then $f \circ g$ tends to $f(l)$ in $F$.
If $g$ is continuous at $f(x)$ and $f$ is continuous at $x$ within $s$, then $g \circ f$ is continuous at $x$ within $s$.
If $f$ is continuous at $a$ and $f$ is an open map, then the filter of neighborhoods of $a$ is mapped to the filter of neighborhoods of $f(a)$.
A function is continuous at a point $x$ if and only if it is continuous from the left and from the right at $x$.
If $f$ and $g$ are continuous functions on a set $A$, then the function $x \mapsto \max(f(x), g(x))$ is continuous on $A$.
If $f$ and $g$ are continuous functions on a set $A$, then the function $h(x) = \min(f(x), g(x))$ is also continuous on $A$.
If $f$ and $g$ are continuous functions, then so is the function $x \mapsto \max(f(x), g(x))$.
If $f$ and $g$ are continuous functions, then so is the function $x \mapsto \min(f(x), g(x))$.
If $f$ and $g$ are continuous functions from a topological space to a Hausdorff space, then the set of points where $f$ and $g$ differ is open.
If $f$ and $g$ are continuous functions from $\mathbb{R}$ to $\mathbb{R}$, then the set $\{x \in \mathbb{R} \mid f(x) = g(x)\}$ is closed.
If $f$ and $g$ are continuous real-valued functions, then the set $\{x \mid f(x) < g(x)\}$ is open.
The set of points $x$ such that $f(x) \leq g(x)$ is closed.
A set $s$ is compact if for every collection of open sets whose union contains $s$, there is a finite subcollection whose union also contains $s$.
The empty set is compact.
If $S$ is a compact set and $\T$ is a collection of open sets such that $S \subseteq \bigcup \T$, then there exists a finite subcollection $\T'$ of $\T$ such that $S \subseteq \bigcup \T'$.
Suppose $S$ is a compact set and $f$ is a collection of open sets such that $S \subseteq \bigcup_{c \in C} f(c)$. Then there exists a finite subset $C'$ of $C$ such that $S \subseteq \bigcup_{c \in C'} f(c)$.
The intersection of a compact set and a closed set is compact.
If $S$ is compact and $T$ is open, then $S - T$ is compact.
The function $A \mapsto -A$ is injective on sets of real numbers.
A set $U$ is compact if and only if for every collection of closed sets $A$, if every finite subcollection of $A$ has nonempty intersection, then $A$ has nonempty intersection.
If $S$ is compact and each $T \in F$ is closed, then $S \cap \bigcap F \neq \emptyset$.
If $s$ is compact and each $f_i$ is closed, then $\bigcap_{i \in I} f_i$ is nonempty.
In a T2 space, compact sets are closed.
If $f$ is a continuous function from a compact set $S$ to a topological space $X$, then the image $f(S)$ is compact.
If $f$ is a continuous function from a compact set $S$ to a Hausdorff space $Y$, and $g$ is a function from $Y$ to $S$ such that $g(f(x)) = x$ for all $x \in S$, then $g$ is continuous.
If $f$ is a continuous injective map from a compact space to a Hausdorff space, then the inverse of $f$ is continuous.
If $S$ is a compact subset of a linearly ordered topological space, then $S$ has a maximum element.
If $S$ is a nonempty compact subset of a linearly ordered topological space, then there exists an element $s \in S$ such that $s \leq t$ for all $t \in S$.
If $f$ is a continuous function on a compact set $S$ and $S$ is nonempty, then $f$ attains its supremum on $S$.
If $f$ is a continuous function on a compact set $S$ and $S$ is nonempty, then $f$ attains its infimum.
If $U$ is a subset of a topological space $X$ such that for any two open sets $A$ and $B$ in $X$ with $A \cap U \neq \emptyset$, $B \cap U \neq \emptyset$, and $A \cap B \cap U = \emptyset$, we have $U \subseteq A \cup B$, then $U$ is connected.
The empty set is connected.
A singleton set is connected.
If $A$ is connected, $U$ and $V$ are open, $U \cap V \cap A = \emptyset$, and $A \subseteq U \cup V$, then $U \cap A = \emptyset$ or $V \cap A = \emptyset$.
A set $S$ is connected if and only if it is not the union of two disjoint closed sets.
If $s$ is a connected set and $A$ and $B$ are closed sets such that $s \subseteq A \cup B$ and $A \cap B \cap s = \emptyset$, then either $A \cap s = \emptyset$ or $B \cap s = \emptyset$.
If each member of a collection of connected sets has nonempty intersection, then the union of the collection is connected.
If $s$ and $t$ are connected sets with nonempty intersection, then $s \cup t$ is connected.
If $s \subseteq t \subseteq u$ and $s$ is open, $t$ is closed, $u$ is connected, and $t - s$ is connected, then $u - s$ is connected.
A set $S$ is connected if and only if every continuous function $f: S \to \{0,1\}$ is constant.
If $S$ is a connected set and $P$ is a continuous function on $S$, then $P$ is constant.
If every continuous function on $S$ is constant, then $S$ is connected.
If $A$ is a connected set and $f$ is a function that is locally constant on $A$, then $f$ is constant on $A$.
If $U$ is a connected subset of a linearly ordered topological space, and $x, y \in U$ with $x \leq z \leq y$, then $z \<in> U$.
If $S$ is a connected subset of a linearly ordered set, and $x$ is not in $S$, then either $S$ is bounded above and $x \geq y$ for all $y \in S$, or $S$ is bounded below and $x \leq y$ for all $y \in S$.
If $f$ is a continuous function from a connected set $S$ to a topological space $X$, then $f(S)$ is connected.
If $A$ is an open set and $x$ is a lower bound for $A$, then the infimum of $A$ is not in $A$.
If $A$ is an open set and $x$ is an upper bound for $A$, then the supremum of $A$ is not in $A$.
If $U$ is a subset of a linear continuum such that for all $x, y, z \in U$ with $x \leq z \leq y$, we have $z \in U$, then $U$ is connected.
A subset of a linear continuum is connected if and only if it is an interval.
The real line is connected.
The set of all real numbers greater than $a$ is connected.
The interval $[a, \infty)$ is connected.
The interval $(-\infty, a)$ is connected.
The interval $[-\infty, a]$ is connected.
The open interval $(a,b)$ is connected.
The interval $(a,b)$ is connected.
The interval $(a, b)$ is connected.
The interval $[a,b]$ is connected.
If $A$ is a connected set and $a, b \in A$, then the open interval $(a, b)$ is contained in $A$.
If $A$ is a connected set and $a, b \in A$, then the interval $[a, b]$ is contained in $A$.
If $f$ is a continuous function on a closed interval $[a,b]$ and $y$ is a number between $f(a)$ and $f(b)$, then there is a number $x$ between $a$ and $b$ such that $f(x) = y$.
If $f$ is a continuous function on the closed interval $[a, b]$ and $f(a) \leq y \leq f(b)$, then there exists $x \in [a, b]$ such that $f(x) = y$.
If $f$ is continuous on the interval $[a,b]$ and $f(a) \leq y \leq f(b)$, then there exists $x \in [a,b]$ such that $f(x) = y$.
If $f$ is a continuous function on the interval $[a,b]$ and $f(a) \leq y \leq f(b)$, then there exists $x \in [a,b]$ such that $f(x) = y$.
If $f$ is a continuous injective function on an interval $[a,b]$, then $f(a) < f(x) < f(b)$ or $f(b) < f(x) < f(a)$.
If $f$ is a monotone function and continuous at the supremum of a nonempty bounded set $S$, then $f$ is continuous at the supremum of $S$.
If $f$ is an antimonotone function and $f$ is continuous at the supremum of a nonempty bounded set $S$, then $f$ attains its minimum at the supremum of $S$.
If $f$ is a monotone function and $f$ is continuous at the right of the infimum of a nonempty set $S$ that is bounded below, then $f$ is continuous at the infimum of $S$.
If $f$ is an antimonotone function and $f$ is continuous at the right of the infimum of a nonempty set $S$ that is bounded below, then $f$ applied to the infimum of $S$ is equal to the supremum of $f$ applied to the elements of $S$.
The uniformity is not executable.
The uniformity on a set is never empty.
If $E$ is eventually true in the uniformity, then the predicate $(x, y, z) \mapsto y = y' \implies E(x, z)$ is eventually true in the product uniformity.
If $E$ is eventually in the uniformity, then there exists $D$ in the uniformity such that $D(x,y)$ and $D(y,z)$ implies $E(x,z)$.
For any point $x$ and any property $P$, the following are equivalent: $P$ holds for all points $y$ sufficiently close to $x$. For all $\epsilon > 0$, there exists $\delta > 0$ such that if $d(x, x') < \delta$, then $d(y, y') < \epsilon$ implies $P(y')$.

If $f$ is continuous at $a$ from the right, continuous at $b$ from the left, and continuous on $(a,b)$, then $f$ is continuous on $[a,b]$.

If $f$ is continuous on the closed interval $[a, b]$, then $f$ is continuous at $a$ from the right and at $b$ from the left.

A function from a discrete space to any space is continuous.

The constant function $f(x) = \bot$ is continuous.

If a net has a trivial limit, then any function on the net is continuous.

A function $f$ is continuous at $x$ within $S$ if and only if the limit of $f$ at $x$ within $S$ exists and is equal to $f(x)$.

A function $f$ is continuous at $x$ within $S$ if and only if for every open set $B$ containing $f(x)$, there exists an open set $A$ containing $x$ such that $f(y) \in B$ for all $y \in S \cap A$.

If $f$ is continuous at $x$ within $s$ and $g$ is continuous at $f(x)$ within $f(s)$, then $g \circ f$ is continuous at $x$ within $s$.

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

A function $f$ is continuous at $x$ if and only if $f$ converges to $f(x)$ as $x$ approaches $x$.

The identity function is continuous.

The constant function $x \mapsto c$ is continuous.

A function $f$ is continuous on a set $S$ if and only if it is continuous at every point of $S$.

The function $f$ is continuous at every point $x$ in the discrete topology.

A function $f$ is continuous at $a$ if and only if $f$ has a limit at $a$.

If $f$ is continuous at $x$, then $f$ is defined at $x$.

If $f$ and $g$ are two functions that agree on a neighborhood of $x$, then $f$ is continuous at $x$ if and only if $g$ is continuous at $x$.

If $f$ is continuous at $x$, then $f$ is continuous at $x$ within $s$.

If $f$ is continuous on an open set $S$, then $f$ is continuous at every point of $S$.

If $f$ is continuous at $a$ within $A$, then $f$ is continuous at $a$.

If $f$ is continuous at every point of $s$, then $f$ is continuous on $s$.

If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$.

If $g$ is continuous at $l$ and $f$ tends to $l$, then $g \circ f$ tends to $g(l)$.

If $f$ is continuous on $s$, $g$ tends to $l$ in $F$, $l$ is in $s$, and $g$ is eventually in $s$, then $f \circ g$ tends to $f(l)$ in $F$.

If $g$ is continuous at $f(x)$ and $f$ is continuous at $x$ within $s$, then $g \circ f$ is continuous at $x$ within $s$.

If $f$ is continuous at $a$ and $f$ is an open map, then the filter of neighborhoods of $a$ is mapped to the filter of neighborhoods of $f(a)$.

A function is continuous at a point $x$ if and only if it is continuous from the left and from the right at $x$.

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

If $f$ and $g$ are continuous functions on a set $A$, then the function $h(x) = \min(f(x), g(x))$ is also continuous on $A$.

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

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

If $f$ and $g$ are continuous functions from a topological space to a Hausdorff space, then the set of points where $f$ and $g$ differ is open.

If $f$ and $g$ are continuous functions from $\mathbb{R}$ to $\mathbb{R}$, then the set $\{x \in \mathbb{R} \mid f(x) = g(x)\}$ is closed.

If $f$ and $g$ are continuous real-valued functions, then the set $\{x \mid f(x) < g(x)\}$ is open.

The set of points $x$ such that $f(x) \leq g(x)$ is closed.

A set $s$ is compact if for every collection of open sets whose union contains $s$, there is a finite subcollection whose union also contains $s$.

The empty set is compact.

If $S$ is a compact set and $\T$ is a collection of open sets such that $S \subseteq \bigcup \T$, then there exists a finite subcollection $\T'$ of $\T$ such that $S \subseteq \bigcup \T'$.

Suppose $S$ is a compact set and $f$ is a collection of open sets such that $S \subseteq \bigcup_{c \in C} f(c)$. Then there exists a finite subset $C'$ of $C$ such that $S \subseteq \bigcup_{c \in C'} f(c)$.

The intersection of a compact set and a closed set is compact.

If $S$ is compact and $T$ is open, then $S - T$ is compact.

The function $A \mapsto -A$ is injective on sets of real numbers.

A set $U$ is compact if and only if for every collection of closed sets $A$, if every finite subcollection of $A$ has nonempty intersection, then $A$ has nonempty intersection.

If $S$ is compact and each $T \in F$ is closed, then $S \cap \bigcap F \neq \emptyset$.

If $s$ is compact and each $f_i$ is closed, then $\bigcap_{i \in I} f_i$ is nonempty.

In a T2 space, compact sets are closed.

If $f$ is a continuous function from a compact set $S$ to a topological space $X$, then the image $f(S)$ is compact.

If $f$ is a continuous function from a compact set $S$ to a Hausdorff space $Y$, and $g$ is a function from $Y$ to $S$ such that $g(f(x)) = x$ for all $x \in S$, then $g$ is continuous.

If $f$ is a continuous injective map from a compact space to a Hausdorff space, then the inverse of $f$ is continuous.

If $S$ is a compact subset of a linearly ordered topological space, then $S$ has a maximum element.

If $S$ is a nonempty compact subset of a linearly ordered topological space, then there exists an element $s \in S$ such that $s \leq t$ for all $t \in S$.

If $f$ is a continuous function on a compact set $S$ and $S$ is nonempty, then $f$ attains its supremum on $S$.

If $f$ is a continuous function on a compact set $S$ and $S$ is nonempty, then $f$ attains its infimum.

If $U$ is a subset of a topological space $X$ such that for any two open sets $A$ and $B$ in $X$ with $A \cap U \neq \emptyset$, $B \cap U \neq \emptyset$, and $A \cap B \cap U = \emptyset$, we have $U \subseteq A \cup B$, then $U$ is connected.

The empty set is connected.

A singleton set is connected.

If $A$ is connected, $U$ and $V$ are open, $U \cap V \cap A = \emptyset$, and $A \subseteq U \cup V$, then $U \cap A = \emptyset$ or $V \cap A = \emptyset$.

A set $S$ is connected if and only if it is not the union of two disjoint closed sets.

If $s$ is a connected set and $A$ and $B$ are closed sets such that $s \subseteq A \cup B$ and $A \cap B \cap s = \emptyset$, then either $A \cap s = \emptyset$ or $B \cap s = \emptyset$.

If each member of a collection of connected sets has nonempty intersection, then the union of the collection is connected.

If $s$ and $t$ are connected sets with nonempty intersection, then $s \cup t$ is connected.

If $s \subseteq t \subseteq u$ and $s$ is open, $t$ is closed, $u$ is connected, and $t - s$ is connected, then $u - s$ is connected.

A set $S$ is connected if and only if every continuous function $f: S \to \{0,1\}$ is constant.

If $S$ is a connected set and $P$ is a continuous function on $S$, then $P$ is constant.

If every continuous function on $S$ is constant, then $S$ is connected.

If $A$ is a connected set and $f$ is a function that is locally constant on $A$, then $f$ is constant on $A$.

If $U$ is a connected subset of a linearly ordered topological space, and $x, y \in U$ with $x \leq z \leq y$, then $z \<in> U$.

If $S$ is a connected subset of a linearly ordered set, and $x$ is not in $S$, then either $S$ is bounded above and $x \geq y$ for all $y \in S$, or $S$ is bounded below and $x \leq y$ for all $y \in S$.

If $f$ is a continuous function from a connected set $S$ to a topological space $X$, then $f(S)$ is connected.

If $A$ is an open set and $x$ is a lower bound for $A$, then the infimum of $A$ is not in $A$.

If $A$ is an open set and $x$ is an upper bound for $A$, then the supremum of $A$ is not in $A$.

If $U$ is a subset of a linear continuum such that for all $x, y, z \in U$ with $x \leq z \leq y$, we have $z \in U$, then $U$ is connected.

A subset of a linear continuum is connected if and only if it is an interval.

The real line is connected.

The set of all real numbers greater than $a$ is connected.

The interval $[a, \infty)$ is connected.

The interval $(-\infty, a)$ is connected.

The interval $[-\infty, a]$ is connected.

The open interval $(a,b)$ is connected.

The interval $(a,b)$ is connected.

The interval $(a, b)$ is connected.

The interval $[a,b]$ is connected.

If $A$ is a connected set and $a, b \in A$, then the open interval $(a, b)$ is contained in $A$.

If $A$ is a connected set and $a, b \in A$, then the interval $[a, b]$ is contained in $A$.

If $f$ is a continuous function on a closed interval $[a,b]$ and $y$ is a number between $f(a)$ and $f(b)$, then there is a number $x$ between $a$ and $b$ such that $f(x) = y$.

If $f$ is a continuous function on the closed interval $[a, b]$ and $f(a) \leq y \leq f(b)$, then there exists $x \in [a, b]$ such that $f(x) = y$.

If $f$ is continuous on the interval $[a,b]$ and $f(a) \leq y \leq f(b)$, then there exists $x \in [a,b]$ such that $f(x) = y$.

If $f$ is a continuous function on the interval $[a,b]$ and $f(a) \leq y \leq f(b)$, then there exists $x \in [a,b]$ such that $f(x) = y$.

If $f$ is a continuous injective function on an interval $[a,b]$, then $f(a) < f(x) < f(b)$ or $f(b) < f(x) < f(a)$.

If $f$ is a monotone function and continuous at the supremum of a nonempty bounded set $S$, then $f$ is continuous at the supremum of $S$.

If $f$ is an antimonotone function and $f$ is continuous at the supremum of a nonempty bounded set $S$, then $f$ attains its minimum at the supremum of $S$.

If $f$ is a monotone function and $f$ is continuous at the right of the infimum of a nonempty set $S$ that is bounded below, then $f$ is continuous at the infimum of $S$.

If $f$ is an antimonotone function and $f$ is continuous at the right of the infimum of a nonempty set $S$ that is bounded below, then $f$ applied to the infimum of $S$ is equal to the supremum of $f$ applied to the elements of $S$.

The uniformity is not executable.

The uniformity on a set is never empty.

If $E$ is eventually true in the uniformity, then the predicate $(x, y, z) \mapsto y = y' \implies E(x, z)$ is eventually true in the product uniformity.

If $E$ is eventually in the uniformity, then there exists $D$ in the uniformity such that $D(x,y)$ and $D(y,z)$ implies $E(x,z)$.

For any point $x$ and any property $P$, the following are equivalent: $P$ holds for all points $y$ sufficiently close to $x$. For all $\epsilon > 0$, there exists $\delta > 0$ such that if $d(x, x') < \delta$, then $d(y, y') < \epsilon$ implies $P(y')$.