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The difference between a closed ball and a sphere is an open ball. |
If $d \leq e$, then the ball of radius $d$ centered at $x$ is contained in the ball of radius $e$ centered at $x$. |
If $d \leq e$, then the closed ball of radius $d$ centered at $x$ is contained in the closed ball of radius $e$ centered at $x$. |
If $x$ is in the ball of radius $e$ around $y$, and $e \leq f$, then $x$ is in the ball of radius $f$ around $y$. |
If $x$ is in the closed ball of radius $e$ centered at $y$, and $e \leq f$, then $x$ is in the closed ball of radius $f$ centered at $y$. |
If $y$ is in the closed ball of radius $b$ centered at $z$, and $x$ is in the closed ball of radius $a$ centered at $y$, then $x$ is in the closed ball of radius $b + a$ centered at $z$. |
The ball of radius $max(r,s)$ around $a$ is the union of the balls of radius $r$ and $s$ around $a$. |
The intersection of two balls is a ball. |
The closed ball of radius $\max(r,s)$ centered at $a$ is equal to the union of the closed balls of radius $r$ and $s$ centered at $a$. |
The intersection of two closed balls is the smallest closed ball that contains both of them. |
The boundary of a ball is the sphere. |
The open ball of radius $e$ around $x$ is open. |
A set $S$ is open if and only if for every $x \in S$, there exists an $e > 0$ such that the ball of radius $e$ centered at $x$ is contained in $S$. |
If for every $x \in S$, there exists an open ball around $x$ that is contained in $S$, then $S$ is open. |
If $S$ is an open set and $x \in S$, then there exists an $\epsilon > 0$ such that $B(x, \epsilon) \subseteq S$. |
A set $S$ is open if and only if for every $x \in S$, there exists an $e > 0$ such that the ball of radius $e$ centered at $x$ is contained in $S$. |
The ball of radius $e$ around $x$ is empty if and only if $e \leq 0$. |
If $e \leq 0$, then the ball of radius $e$ around $x$ is empty. |
The closed ball of radius $e$ centered at $x$ is closed. |
A set $S$ is open if and only if for every $x \in S$, there exists an $\epsilon > 0$ such that the open ball of radius $\epsilon$ centered at $x$ is contained in $S$. |
A set is open if and only if for every point in the set, there exists a ball around that point that is contained in the set. |
For any $d > 0$, there exists a neighborhood of $z$ such that all points in that neighborhood are within distance $d$ of $z$. |
If $d > 0$, then there exists a neighborhood of $z$ contained in $A$ such that all points in this neighborhood are in the ball of radius $d$ around $z$. |
If $d > 0$, then there exists a point $t$ in the ball of radius $d$ around $z$ such that $t \neq z$ and $t \in A$. |
If $e > 0$ and $|x - y| < e$, then the filter at $y$ within the ball of radius $e$ centered at $x$ is the same as the filter at $y$. |
The interval $[a, b]$ is equal to the closed ball of radius $(b - a)/2$ centered at $(a + b)/2$. |
The closed ball of radius $b$ centered at $a$ is equal to the closed interval $[a - b, a + b]$. |
The open interval $(a, b)$ is equal to the open ball of radius $(b - a)/2$ centered at $(a + b)/2$. |
The ball of radius $b$ centered at $a$ is equal to the open interval $(a - b, a + b)$. |
The interior of a ball is the ball itself. |
The closed ball of radius $e$ around $x$ is empty if and only if $e < 0$. |
If $e < 0$, then the closed ball of radius $e$ centered at $x$ is empty. |
If $e = 0$, then the closed ball of radius $e$ centered at $x$ is just the point $x$. |
If $d \geq 1$, then the ball of radius $e/d$ centered at $x$ is contained in the ball of radius $e$ centered at $x$. |
If $x$ is a point in a metric space, and $e$ is a positive real number, then the ball of radius $e/n$ around $x$ is contained in the ball of radius $e$ around $x$. |
If $d \geq 1$, then the ball of radius $e/d$ centered at $x$ is contained in the ball of radius $e$ centered at $x$. |
The ball of radius $e/n$ centered at $x$ is contained in the ball of radius $e$ centered at $x$. |
If $a \neq 0$, then the image of the closed ball of radius $r$ centered at $c$ under the map $x \mapsto a \cdot x$ is the closed ball of radius $\lvert a \rvert \cdot r$ centered at $a \cdot c$. |
If $a \neq 0$, then the image of the ball of radius $r$ centered at $c$ under the map $x \mapsto a \cdot x$ is the ball of radius $\lvert a \rvert \cdot r$ centered at $a \cdot c$. |
A point $x$ is a limit point of a set $S$ if and only if for every $\epsilon > 0$, there exists a point $x' \in S$ such that $x' \neq x$ and $d(x', x) < \epsilon$. |
A point $x$ is a limit point of a set $S$ if and only if for every $\epsilon > 0$, there exists a point $x' \in S$ such that $x' \neq x$ and $d(x', x) \leq \epsilon$. |
If $x$ is a limit point of the set of limit points of $S$, then $x$ is a limit point of $S$. |
The set of limit points of a set $S$ is closed. |
A point $x$ is a limit point of the closure of a set $S$ if and only if $x$ is a limit point of $S$. |
A point $x$ is a limit point of a set $S$ if and only if every neighborhood of $x$ contains infinitely many points of $S$. |
A point $x$ is a limit point of a set $S$ if and only if for every $\epsilon > 0$, the intersection of $S$ with the closed ball of radius $\epsilon$ centered at $x$ is infinite. |
If $x$ is a point in a perfect metric space, then for every positive real number $r$, there exists a point $a$ such that $a \neq x$ and $d(a,x) < r$. |
For any metric space $X$ and any point $x \in X$, the closed ball of radius $e$ centered at $x$ is equal to $\{x\}$ if and only if $e = 0$. |
If $S$ is a finite set, then there exists a positive real number $e$ such that $S \subseteq B(a, e)$. |
If $S$ is a finite set, then there exists a positive real number $d$ such that for all $x \in S$, if $x \neq a$, then $d \leq \|x - a\|$. |
If $S$ is a discrete set, then $S$ is closed. |
A point $x$ is in the interior of a set $S$ if and only if there exists an open ball around $x$ that is contained in $S$. |
A point $x$ is in the interior of a set $S$ if and only if there exists an $e > 0$ such that the closed ball of radius $e$ centered at $x$ is contained in $S$. |
A point $a$ is in the frontier of a set $S$ if and only if for every $\epsilon > 0$, there exists a point $x \in S$ such that $d(a, x) < \epsilon$ and there exists a point $x$ not in $S$ such that $d(a, x) < \epsilon$. |
A net $f$ converges to $l$ if and only if either $f$ is eventually constant or for every $\epsilon > 0$, there exists $x$ such that $|f(x) - l| < \epsilon$. |
The limit of $f$ at $a$ within $S$ is $l$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in S$, if $0 < |x - a| \leq \delta$, then $|f(x) - l| < \epsilon$. |
A function $f$ converges to $l$ at $a$ within $S$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in S$, if $0 < |x - a| < \delta$, then $|f(x) - l| < \epsilon$. |
If $f$ is a function defined on a set $S$ and $a$ is a point in $S$, then $f$ converges to $l$ at $a$ within $S$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in S$ with $0 < |x - a| < \delta$, we have $|f(x) - l| \leq \epsilon$. |
A function $f$ converges to $l$ at $a$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x$, if $0 < |x - a| < \delta$, then $|f(x) - l| < \epsilon$. |
If $f$ converges to $l$ at $a$ within $S$, and $S$ and $T$ are eventually equal near $a$, then $f$ converges to $l$ at $a$ within $T$. |
A point $x$ is a limit point of a set $S$ if and only if there exists an injective sequence $(x_n)$ in $S$ that converges to $x$. |
If $f$ converges to $l$ and $f$ is eventually within $e$ of $a$, then $l$ is within $e$ of $a$. |
A function $f$ is continuous at $x$ within $S$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x' \in S$, if $|x' - x| < \delta$, then $|f(x') - f(x)| < \epsilon$. |
A function $f$ is continuous at $x$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x'$, if $|x' - x| < \delta$, then $|f(x') - f(x)| < \epsilon$. |
A real-valued function $f$ is continuous from the right at $a$ if and only if for every $\epsilon > 0$, there exists $\delta > 0$ such that $f(a + \delta) - f(a) < \epsilon$. |
If $f$ is nondecreasing, then $f$ is continuous from the left at $a$ if and only if for every $\epsilon > 0$, there exists $\delta > 0$ such that $f(a) - f(a - \delta) < \epsilon$. |
A function $f$ is continuous at $x$ within $s$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $f(B(x, \delta) \cap s) \subseteq B(f(x), \epsilon)$. |
A function $f$ is continuous at $x$ if and only if for every $\epsilon > 0$, there exists $\delta > 0$ such that $f(B(x, \delta)) \subseteq B(f(x), \epsilon)$. |
A function $f$ is continuous on a set $S$ if and only if for every $x \in S$ and every $\epsilon > 0$, there exists a $\delta > 0$ such that for every $x' \in S$, if $|x' - x| < \delta$, then $|f(x') - f(x)| < \epsilon$. |
If $f$ is continuous at $x$ within $s$, then for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x' \in s$ with $|x' - x| < \delta$, we have $|f(x') - f(x)| < \epsilon$. |
If for every $x \in S$ and every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x' \in S$, if $|x' - x| < \delta$, then $|f(x') - f(x)| \leq \epsilon$, then $f$ is continuous on $S$. |
Suppose $f$ is a continuous function defined on a set $S$. If $x \in S$ and $\epsilon > 0$, then there exists $\delta > 0$ such that for all $x' \in S$ with $|x' - x| < \delta$, we have $|f(x') - f(x)| < \epsilon$. |
If $f$ is continuous at $x$ on $S$, and $g$ agrees with $f$ on a neighborhood of $x$ in $S$, then $g$ is continuous at $x$ on $S$. |
A point $x$ is in the closure of a set $S$ if and only if for every $\epsilon > 0$, there exists a point $y \in S$ such that $d(x,y) < \epsilon$. |
A point $x$ is in the closure of a set $S$ if and only if for every $\epsilon > 0$, there exists a point $y \in S$ such that $d(y, x) \leq \epsilon$. |
If $x$ is in the closure of $S$, then there exists a point $y \in S$ such that $x$ is within distance $e$ of $y$. |
If $S$ is a closed set, then $x \in S$ if and only if for every $\epsilon > 0$, there exists $y \in S$ such that $|x - y| < \epsilon$. |
If $S$ is a nonempty set of real numbers that is bounded below, then the infimum of $S$ is in the closure of $S$. |
If $S$ is a nonempty set of real numbers that is bounded above, then the supremum of $S$ is in the closure of $S$. |
A function $f$ has a non-trivial limit at $x$ if and only if for every $\epsilon > 0$, there exists a point $y \in S$ such that $0 < |x - y| < \epsilon$. |
A set $S$ is bounded if and only if there exists a real number $e \geq 0$ and a point $x$ such that $S \subseteq B(x, e)$. |
A set $S$ is bounded if and only if there exists a real number $e$ such that for all $y \in S$, we have $|y - a| \leq e$. |
A set $S$ is bounded if and only if there exists a real number $a$ such that for all $x \in S$, we have $|x| \leq a$. |
A set $X$ is bounded if and only if the set of norms of elements of $X$ is bounded above. |
The set of norms of a set of complex numbers is bounded if and only if the set of complex numbers is bounded. |
If every element of a set $S$ has norm at most $B$, then $S$ is bounded. |
The empty set is bounded. |
If $T$ is a bounded set and $S$ is a subset of $T$, then $S$ is bounded. |
If $S$ is bounded, then the interior of $S$ is bounded. |
If $S$ is a bounded set, then its closure is bounded. |
If the image of the closure of a set $S$ under a function $f$ is bounded, then the image of $S$ under $f$ is bounded. |
The closed ball of radius $e$ centered at $x$ is bounded. |
The ball of radius $e$ around $x$ is bounded. |
A set is bounded if and only if its union with any other set is bounded. |
If $F$ is a finite set of bounded sets, then $\bigcup F$ is bounded. |
If $A$ is a finite set and each $B_x$ is a bounded set, then $\bigcup_{x \in A} B_x$ is bounded. |
A set is bounded if and only if it is bounded after inserting an element. |
If $S$ is a subset of the ball of radius $r$ centered at $x$, then $S$ is bounded. |
If $S$ is a bounded subset of a metric space, then there exists a positive real number $r$ such that $S$ is contained in the open ball of radius $r$ centered at $x$. |
If $S$ is a finite set, then $S$ is bounded. |
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