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

Statement: stringlengths 7 24.3k
If two spaces have the same sets, then the spaces restricted to the same set have the same sets.
The sets of a countable space restricted to a subset are the same as the sets of the countable space restricted to the intersection of the subset with the original space.
The sets of a space restricted to the whole space are the same as the sets of the original space.
The sets of a space restricted to a subset $A$ and then restricted to a subset $B$ are the same as the sets of the space restricted to the intersection of $A$ and $B$.
If $\Omega$ is a measurable set, then a set $A$ is measurable with respect to the restriction of the measure space $M$ to $\Omega$ if and only if $A$ is a subset of $\Omega$ and is measurable with respect to $M$.
If two measurable spaces have the same sets, then the restrictions of those spaces to the same set have the same sets.
If $\Omega$ is a subset of the sample space of a measure space $M$, then the $\sigma$-algebra of the restricted measure space $M\|\Omega$ is equal to the $\sigma$-algebra of the preimage measure space $\Omega \times M$ under the projection map $\pi: \Omega \times M \to M$.
If $S$ is a measurable set, then the set $\{x \in S : P(x)\}$ is measurable if and only if the set $\{x \in X : x \in S \text{ and } P(x)\}$ is measurable.
If $f$ is a measurable function from $M$ to $N$, then $f$ is a measurable function from $M \cap \Omega$ to $N$.
A function $f$ is measurable with respect to the restriction of a measure space $N$ to a set $\Omega$ if and only if $f$ is measurable with respect to $N$ and $f$ is a function from the space of $M$ to $\Omega$.
If $f$ is a measurable function from a measure space $M$ to a measure space $N$, then the restriction of $f$ to a subset $\Omega$ of $N$ is a measurable function from $M$ to the measure space $N$ restricted to $\Omega$.
If $f$ is measurable on each of a countable collection of sets whose union is the whole space, then $f$ is measurable on the whole space.
If $f$ is a measurable function from a measurable space $M$ to a measurable space $N$, then $f$ is measurable on each measurable subset of $M$.
If $P$ is a measurable predicate, then the function $f$ is measurable if and only if the restrictions of $f$ to the sets $\{x \in X : P(x)\}$ and $\{x \in X : \lnot P(x)\}$ are measurable.
If $f$ and $g$ are measurable functions from $M$ to $M'$, and $P$ is a measurable predicate on $M$, then the function $h$ defined by $h(x) = f(x)$ if $P(x)$ is true, and $h(x) = g(x)$ if $P(x)$ is false, is measurable.
If $f$ and $g$ are measurable functions from a measure space $M$ to a measure space $M'$, and $A$ is a measurable subset of $M$, then the function $h$ defined by $h(x) = f(x)$ if $x \<in> A$ and $h(x) = g(x)$ if $x \<notin> A$ is measurable.
If $\Omega$ is a measurable subset of the space of a measure $M$, and $c$ is an element of the space of a measure $N$, then a function $f$ is measurable from the restriction of $M$ to $\Omega$ to $N$ if and only if the function $x \mapsto f(x)$ if $x \in \Omega$ and $c$ otherwise is measurable from $M$ to $N$.
If $x$ is an element of the underlying set of a measure space $M$, then the measure space $M$ restricted to the singleton set $\{x\}$ is isomorphic to the measure space $\{x\}$ with the counting measure.
If $f$ is a measurable function from a space $M$ to a space $N$ and $X$ is a countable subset of $M$, then $f$ is also measurable when restricted to $M \setminus X$.
If $f$ is a measurable function from a measurable space $M$ to a measurable space $N$, and $X$ is a countable subset of $M$ such that $f$ and $g$ agree on $M \setminus X$, then $g$ is measurable.
If $f$ is a measurable function from a measure space $M$ to a countable set $A$, and $A$ is a subset of a countable set $B$, then $f$ is a measurable function from $M$ to $B$.
If $f$ is a sequence of complex numbers and $z$ is a complex number, then $\limsup_{n \to \infty} \sqrt[n]{\|f_n z^n\|} = \limsup_{n \to \infty} \sqrt[n]{\|f_n\|} \|z\|$.
If the sequence $(\sqrt[n]{\\|f_n\\|})$ converges to $l$, then the limit superior of the sequence is also $l$.
If $\lim_{n \to \infty} \sqrt[n]{\|f_n\|} = l$, then $\limsup_{n \to \infty} \sqrt[n]{\|f_n\|} = l$.
If the root test converges, then the series converges.
If the root test for a series $\sum a_n$ gives a limit superior greater than 1, then the series diverges.
The empty set is open.
The union of two open sets is open.
If $B_x$ is open for all $x \in A$, then $\bigcup_{x \in A} B_x$ is open.
If $S$ is a finite set of open sets, then $\bigcap S$ is open.
If $A$ is a finite set and each $B_x$ is open, then $\bigcap_{x \in A} B_x$ is open.
If every point in a set $S$ has an open neighborhood contained in $S$, then $S$ is open.
A set is open if and only if every point in the set is contained in an open set that is contained in the set.
The empty set is closed.
The union of two closed sets is closed.
The whole space is closed.
The intersection of two closed sets is closed.
If $B_x$ is closed for all $x \in A$, then $\bigcap_{x \in A} B_x$ is closed.
The intersection of a collection of closed sets is closed.
The union of a finite collection of closed sets is closed.
The union of a finite collection of closed sets is closed.
A set is open if and only if its complement is closed.
A set $S$ is closed if and only if its complement $-S$ is open.
If $S$ is open and $T$ is closed, then $S - T$ is open.
If $S$ is closed and $T$ is open, then $S - T$ is closed.
The complement of a closed set is open.
The complement of an open set is closed.
If a set is closed, then its complement is open.
If $P$ and $Q$ are open sets, then $P \cap Q$ is an open set.
If $P$ and $Q$ are open sets, then $P \cup Q$ is an open set.
If each set $\{x \mid P_i(x)\}$ is open, then so is the set $\{x \mid \exists i. P_i(x)\}$.
If $P$ is closed and $Q$ is open, then the set of points $x$ such that $P(x) \implies Q(x)$ is open.
The set of all $x$ such that $P$ is open.
If the set of points $x$ such that $P(x)$ is true is open, then the set of points $x$ such that $P(x)$ is false is closed.
The intersection of two closed sets is closed.
If $P$ and $Q$ are closed sets, then $P \cup Q$ is closed.
If $P_i$ is a closed set for each $i$, then $\{x \mid \forall i. P_i(x)\}$ is closed.
If the set of points $x$ such that $P(x)$ is true is open, and the set of points $x$ such that $Q(x)$ is true is closed, then the set of points $x$ such that $P(x) \implies Q(x)$ is true is closed.
The set of all $x$ such that $P$ is closed.
Two points $x$ and $y$ are distinct if and only if there exists an open set $U$ such that $x \in U$ and $y \notin U$.
The singleton set $\{a\}$ is closed.
If $S$ is a closed set, then $S \cup \{a\}$ is closed.
If $S$ is a finite set, then $S$ is closed.
In a $T_2$ space, two points are distinct if and only if there exist disjoint open sets containing them.
In a $T_0$ space, two points are distinct if and only if there is an open set that contains one of them but not the other.
In a perfect space, the whole space is not a singleton.
If $K_i$ generates the topology on $S$ for each $i \in I$, then $\bigcup_{i \in I} K_i$ generates the topology on $S$.
The topology generated by a set of subsets of a set is a topology.
The set of all real numbers greater than $a$ is open.
The set $\{x \in \mathbb{R} \mid x < a\}$ is open.
The set of real numbers between $a$ and $b$ is open.
The set $\{x \in \mathbb{R} \mid x \leq a\}$ is closed.
The set $\{x \in \mathbb{R} \mid x \geq a\}$ is closed.
The closed interval $[a, b]$ is closed.
If $x < y$ in a totally ordered set, then there exist $a$ and $b$ such that $x \in (-\infty, a)$, $y \in (b, \infty)$, and $(-\infty, a) \cap (b, \infty) = \emptyset$.
If $S$ is an open set in a linearly ordered topology, and $x \in S$, then there exists a point $b > x$ such that the interval $(x, b)$ is contained in $S$.
If $S$ is an open set in a linearly ordered topology, and $x \in S$, then there exists a point $b < x$ such that the interval $(b, x)$ is contained in $S$.
If the open sets of a topological space are generated by a set $T$ of subsets, then the neighborhood filter of a point $x$ is the intersection of all sets in $T$ that contain $x$.
A predicate $P$ holds eventually in the neighborhood filter of $a$ if and only if there exists an open set $S$ containing $a$ such that $P$ holds for all $x \in S$.
For any point $x$ and any property $P$, the set of points $y$ such that $P$ holds in a neighborhood of $y$ is a neighborhood of $x$ if and only if $P$ holds in a neighborhood of $x$.
If $s$ is an open set containing $x$, then there is a neighborhood of $x$ contained in $s$.
If a property $P$ holds in a neighborhood of $x$, then $P$ holds at $x$.
The neighborhood filter of a point in a topological space is never empty.
If $x \neq y$, then for every open neighborhood $U$ of $x$, there exists an open neighborhood $V$ of $x$ such that $V \subseteq U$ and $V \cap \{y\} = \emptyset$.
If $x$ is an isolated point, then the neighborhood filter of $x$ is the principal filter generated by $x$.
In a discrete topology, the neighborhood of a point is the singleton set containing that point.
In a discrete topology, the filter of neighborhoods of a point is the empty filter.
In a discrete topology, a function $f$ converges to $y$ if and only if $f$ eventually equals $y$.
The filter at $x$ within $s$ is the intersection of all open sets containing $x$ that are contained in $s$.
If $P$ holds eventually at $a$ within $s$, then $P$ holds eventually in a neighborhood of $a$ if $x \neq a$ and $x \in s$.
If $s \subseteq t$, then the filter of neighborhoods of $x$ in $s$ is finer than the filter of neighborhoods of $x$ in $t$.
If $P$ holds eventually at $a$ within $s$, then there exists an open set $S$ containing $a$ such that $P$ holds for all $x \in S$ with $x \neq a$ and $x \in s$.
If $a$ is in an open set $S$, then the filter of neighbourhoods of $a$ within $S$ is the same as the filter of neighbourhoods of $a$.
If $a$ is in an open set $s$ and $s$ is disjoint from the complement of $s$, then the filter of neighbourhoods of $a$ within $s$ is the same as the filter of neighbourhoods of $a$.
If $a$ is in an open subset $S$ of a topological space $T$, then the filter of neighbourhoods of $a$ within $T$ is the same as the filter of neighbourhoods of $a$ in $S$.
If $x$ is in a set $S$ and $S$ is open, and if $T$ and $U$ are sets such that $T \cap S - \{x\} = U \cap S - \{x\}$, then the filter of neighborhoods of $x$ in $T$ is the same as the filter of neighborhoods of $x$ in $U$.
The filter of neighbourhoods of a point $a$ in an empty set is the empty filter.
The filter of neighbourhoods of $x$ in $S \cup T$ is the supremum of the filters of neighbourhoods of $x$ in $S$ and $T$.
In a topological space, the filter at a point $a$ is the empty filter if and only if the singleton set $\{a\}$ is open.
In a perfect space, the filter at a point is never the empty filter.

If two spaces have the same sets, then the spaces restricted to the same set have the same sets.

The sets of a countable space restricted to a subset are the same as the sets of the countable space restricted to the intersection of the subset with the original space.

The sets of a space restricted to the whole space are the same as the sets of the original space.

The sets of a space restricted to a subset $A$ and then restricted to a subset $B$ are the same as the sets of the space restricted to the intersection of $A$ and $B$.

If $\Omega$ is a measurable set, then a set $A$ is measurable with respect to the restriction of the measure space $M$ to $\Omega$ if and only if $A$ is a subset of $\Omega$ and is measurable with respect to $M$.

If two measurable spaces have the same sets, then the restrictions of those spaces to the same set have the same sets.

If $\Omega$ is a subset of the sample space of a measure space $M$, then the $\sigma$-algebra of the restricted measure space $M|\Omega$ is equal to the $\sigma$-algebra of the preimage measure space $\Omega \times M$ under the projection map $\pi: \Omega \times M \to M$.

If $S$ is a measurable set, then the set $\{x \in S : P(x)\}$ is measurable if and only if the set $\{x \in X : x \in S \text{ and } P(x)\}$ is measurable.

If $f$ is a measurable function from $M$ to $N$, then $f$ is a measurable function from $M \cap \Omega$ to $N$.

A function $f$ is measurable with respect to the restriction of a measure space $N$ to a set $\Omega$ if and only if $f$ is measurable with respect to $N$ and $f$ is a function from the space of $M$ to $\Omega$.

If $f$ is a measurable function from a measure space $M$ to a measure space $N$, then the restriction of $f$ to a subset $\Omega$ of $N$ is a measurable function from $M$ to the measure space $N$ restricted to $\Omega$.

If $f$ is measurable on each of a countable collection of sets whose union is the whole space, then $f$ is measurable on the whole space.

If $f$ is a measurable function from a measurable space $M$ to a measurable space $N$, then $f$ is measurable on each measurable subset of $M$.

If $P$ is a measurable predicate, then the function $f$ is measurable if and only if the restrictions of $f$ to the sets $\{x \in X : P(x)\}$ and $\{x \in X : \lnot P(x)\}$ are measurable.

If $f$ and $g$ are measurable functions from $M$ to $M'$, and $P$ is a measurable predicate on $M$, then the function $h$ defined by $h(x) = f(x)$ if $P(x)$ is true, and $h(x) = g(x)$ if $P(x)$ is false, is measurable.

If $f$ and $g$ are measurable functions from a measure space $M$ to a measure space $M'$, and $A$ is a measurable subset of $M$, then the function $h$ defined by $h(x) = f(x)$ if $x \<in> A$ and $h(x) = g(x)$ if $x \<notin> A$ is measurable.

If $\Omega$ is a measurable subset of the space of a measure $M$, and $c$ is an element of the space of a measure $N$, then a function $f$ is measurable from the restriction of $M$ to $\Omega$ to $N$ if and only if the function $x \mapsto f(x)$ if $x \in \Omega$ and $c$ otherwise is measurable from $M$ to $N$.

If $x$ is an element of the underlying set of a measure space $M$, then the measure space $M$ restricted to the singleton set $\{x\}$ is isomorphic to the measure space $\{x\}$ with the counting measure.

If $f$ is a measurable function from a space $M$ to a space $N$ and $X$ is a countable subset of $M$, then $f$ is also measurable when restricted to $M \setminus X$.

If $f$ is a measurable function from a measurable space $M$ to a measurable space $N$, and $X$ is a countable subset of $M$ such that $f$ and $g$ agree on $M \setminus X$, then $g$ is measurable.

If $f$ is a measurable function from a measure space $M$ to a countable set $A$, and $A$ is a subset of a countable set $B$, then $f$ is a measurable function from $M$ to $B$.

If $f$ is a sequence of complex numbers and $z$ is a complex number, then $\limsup_{n \to \infty} \sqrt[n]{|f_n z^n|} = \limsup_{n \to \infty} \sqrt[n]{|f_n|} |z|$.

If the sequence $(\sqrt[n]{\|f_n\|})$ converges to $l$, then the limit superior of the sequence is also $l$.

If $\lim_{n \to \infty} \sqrt[n]{|f_n|} = l$, then $\limsup_{n \to \infty} \sqrt[n]{|f_n|} = l$.

If the root test converges, then the series converges.

If the root test for a series $\sum a_n$ gives a limit superior greater than 1, then the series diverges.

The empty set is open.

The union of two open sets is open.

If $B_x$ is open for all $x \in A$, then $\bigcup_{x \in A} B_x$ is open.

If $S$ is a finite set of open sets, then $\bigcap S$ is open.

If $A$ is a finite set and each $B_x$ is open, then $\bigcap_{x \in A} B_x$ is open.

If every point in a set $S$ has an open neighborhood contained in $S$, then $S$ is open.

A set is open if and only if every point in the set is contained in an open set that is contained in the set.

The empty set is closed.

The union of two closed sets is closed.

The whole space is closed.

The intersection of two closed sets is closed.

If $B_x$ is closed for all $x \in A$, then $\bigcap_{x \in A} B_x$ is closed.

The intersection of a collection of closed sets is closed.

The union of a finite collection of closed sets is closed.

A set is open if and only if its complement is closed.

A set $S$ is closed if and only if its complement $-S$ is open.

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

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

The complement of a closed set is open.

The complement of an open set is closed.

If a set is closed, then its complement is open.

If $P$ and $Q$ are open sets, then $P \cap Q$ is an open set.

If $P$ and $Q$ are open sets, then $P \cup Q$ is an open set.

If each set $\{x \mid P_i(x)\}$ is open, then so is the set $\{x \mid \exists i. P_i(x)\}$.

If $P$ is closed and $Q$ is open, then the set of points $x$ such that $P(x) \implies Q(x)$ is open.

The set of all $x$ such that $P$ is open.

If the set of points $x$ such that $P(x)$ is true is open, then the set of points $x$ such that $P(x)$ is false is closed.

The intersection of two closed sets is closed.

If $P$ and $Q$ are closed sets, then $P \cup Q$ is closed.

If $P_i$ is a closed set for each $i$, then $\{x \mid \forall i. P_i(x)\}$ is closed.

If the set of points $x$ such that $P(x)$ is true is open, and the set of points $x$ such that $Q(x)$ is true is closed, then the set of points $x$ such that $P(x) \implies Q(x)$ is true is closed.

The set of all $x$ such that $P$ is closed.

Two points $x$ and $y$ are distinct if and only if there exists an open set $U$ such that $x \in U$ and $y \notin U$.

The singleton set $\{a\}$ is closed.

If $S$ is a closed set, then $S \cup \{a\}$ is closed.

If $S$ is a finite set, then $S$ is closed.

In a $T_2$ space, two points are distinct if and only if there exist disjoint open sets containing them.

In a $T_0$ space, two points are distinct if and only if there is an open set that contains one of them but not the other.

In a perfect space, the whole space is not a singleton.

If $K_i$ generates the topology on $S$ for each $i \in I$, then $\bigcup_{i \in I} K_i$ generates the topology on $S$.

The topology generated by a set of subsets of a set is a topology.

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

The set $\{x \in \mathbb{R} \mid x < a\}$ is open.

The set of real numbers between $a$ and $b$ is open.

The set $\{x \in \mathbb{R} \mid x \leq a\}$ is closed.

The set $\{x \in \mathbb{R} \mid x \geq a\}$ is closed.

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

If $x < y$ in a totally ordered set, then there exist $a$ and $b$ such that $x \in (-\infty, a)$, $y \in (b, \infty)$, and $(-\infty, a) \cap (b, \infty) = \emptyset$.

If $S$ is an open set in a linearly ordered topology, and $x \in S$, then there exists a point $b > x$ such that the interval $(x, b)$ is contained in $S$.

If $S$ is an open set in a linearly ordered topology, and $x \in S$, then there exists a point $b < x$ such that the interval $(b, x)$ is contained in $S$.

If the open sets of a topological space are generated by a set $T$ of subsets, then the neighborhood filter of a point $x$ is the intersection of all sets in $T$ that contain $x$.

A predicate $P$ holds eventually in the neighborhood filter of $a$ if and only if there exists an open set $S$ containing $a$ such that $P$ holds for all $x \in S$.

For any point $x$ and any property $P$, the set of points $y$ such that $P$ holds in a neighborhood of $y$ is a neighborhood of $x$ if and only if $P$ holds in a neighborhood of $x$.

If $s$ is an open set containing $x$, then there is a neighborhood of $x$ contained in $s$.

If a property $P$ holds in a neighborhood of $x$, then $P$ holds at $x$.

The neighborhood filter of a point in a topological space is never empty.

If $x \neq y$, then for every open neighborhood $U$ of $x$, there exists an open neighborhood $V$ of $x$ such that $V \subseteq U$ and $V \cap \{y\} = \emptyset$.

If $x$ is an isolated point, then the neighborhood filter of $x$ is the principal filter generated by $x$.

In a discrete topology, the neighborhood of a point is the singleton set containing that point.

In a discrete topology, the filter of neighborhoods of a point is the empty filter.

In a discrete topology, a function $f$ converges to $y$ if and only if $f$ eventually equals $y$.

The filter at $x$ within $s$ is the intersection of all open sets containing $x$ that are contained in $s$.

If $P$ holds eventually at $a$ within $s$, then $P$ holds eventually in a neighborhood of $a$ if $x \neq a$ and $x \in s$.

If $s \subseteq t$, then the filter of neighborhoods of $x$ in $s$ is finer than the filter of neighborhoods of $x$ in $t$.

If $P$ holds eventually at $a$ within $s$, then there exists an open set $S$ containing $a$ such that $P$ holds for all $x \in S$ with $x \neq a$ and $x \in s$.

If $a$ is in an open set $S$, then the filter of neighbourhoods of $a$ within $S$ is the same as the filter of neighbourhoods of $a$.

If $a$ is in an open set $s$ and $s$ is disjoint from the complement of $s$, then the filter of neighbourhoods of $a$ within $s$ is the same as the filter of neighbourhoods of $a$.

If $a$ is in an open subset $S$ of a topological space $T$, then the filter of neighbourhoods of $a$ within $T$ is the same as the filter of neighbourhoods of $a$ in $S$.

If $x$ is in a set $S$ and $S$ is open, and if $T$ and $U$ are sets such that $T \cap S - \{x\} = U \cap S - \{x\}$, then the filter of neighborhoods of $x$ in $T$ is the same as the filter of neighborhoods of $x$ in $U$.

The filter of neighbourhoods of a point $a$ in an empty set is the empty filter.

The filter of neighbourhoods of $x$ in $S \cup T$ is the supremum of the filters of neighbourhoods of $x$ in $S$ and $T$.

In a topological space, the filter at a point $a$ is the empty filter if and only if the singleton set $\{a\}$ is open.

In a perfect space, the filter at a point is never the empty filter.