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If $f$ is a continuous injective map from a compact space $S$ to a Hausdorff space $T$, then $S$ and $T$ are homeomorphic. |
If two topological spaces are homeomorphic, then they are compact if and only if they are compact. |
If $a < b$, then $a$ is a limit point of the open interval $(a, b)$. |
If $a < b$, then $b$ is a limit point of the open interval $(a,b)$. |
If $a < b$, then the closure of the open interval $(a, b)$ is the closed interval $[a, b]$. |
The closure of the set of all real numbers greater than $a$ is the set of all real numbers greater than or equal to $a$. |
The closure of the set of all real numbers less than $b$ is the set of all real numbers less than or equal to $b$. |
If $a < b$, then the closure of the open interval $(a, b)$ is the closed interval $[a, b]$. |
If $a < b$, then the closure of the interval $(a, b)$ is the interval $[a, b]$. |
The zero polynomial is not irreducible. |
If $p$ is an irreducible polynomial, then $p$ does not divide $1$. |
The polynomial $1$ is not irreducible. |
If $p$ is a nonzero polynomial that is not a unit, and if $p$ cannot be factored into two nonunits, then $p$ is irreducible. |
If $p$ is irreducible and $p = ab$, then either $a$ divides $1$ or $b$ divides $1$. |
If $b$ is irreducible and $a$ divides $b$ and $a$ is not a unit, then $a$ is irreducible. |
The element $0$ is not a prime element. |
A prime element is not a unit. |
If $p$ is a nonzero element of a commutative ring such that $p$ does not divide $1$, and if $p$ divides the product of any two elements, then $p$ divides one of the elements, then $p$ is a prime element. |
If $p$ is a prime element and $p$ divides $ab$, then $p$ divides $a$ or $p$ divides $b$. |
If $p$ is a prime element, then $p$ divides $ab$ if and only if $p$ divides $a$ or $p$ divides $b$. |
The number 1 is not a prime element. |
If $p$ is a prime element, then $p \neq 0$. |
If $p$ is a prime element and $p$ divides $x^n$, then $p$ divides $x$. |
If $p$ is a prime element and $n > 0$, then $p$ divides $x^n$ if and only if $p$ divides $x$. |
If $x$ is a prime element, then $x \neq 0$. |
If $x$ is a prime element, then $x \neq 1$. |
If two elements of a normalization semidom are equal after normalization, then they are irreducible if and only if the other is irreducible. |
If two elements of a normalization semidomain are equal, then they are associates. |
If $a$ and $b$ are associated elements in a normalization semidom, then there exists a unit $u$ such that $b = ua$. |
If $x$ is a normal element of a normalization semidomain, then $x^n$ is also a normal element. |
If $p$ is a prime element, then $p$ is irreducible. |
If $x$ is a unit and $p$ is irreducible, then $p$ does not divide $x$. |
If $x$ is a unit and $p$ is a prime element, then $p$ does not divide $x$. |
If $p$ is a prime element and $q$ divides $p$, then $q$ is a prime element. |
If $a$ is irreducible and $b$ divides $a$, then either $a$ divides $b$ or $b$ is a unit. |
If $a$ is not a unit and if every divisor of $a$ is either a unit or a multiple of $a$, then $a$ is irreducible. |
An element $x$ of a commutative ring is irreducible if and only if $x$ is not zero, $x$ is not a unit, and for every $b$ that divides $x$, either $x$ divides $b$ or $b$ is a unit. |
If $a$ and $b$ are non-units and $ab$ is a prime element, then $a$ or $b$ is a unit. |
If $p$ is a prime element and $a$ divides $p$ and $a$ is not a unit, then $p$ divides $a$. |
If $p$ is a prime element and $p$ divides the product of the elements of a multiset $A$, then there exists an element $a \in A$ such that $p$ divides $a$. |
The modulus of a complex number is less than or equal to the modulus of the sum of that number and another complex number plus the modulus of the second complex number. |
For any polynomial $p$ and any real number $r$, there exists a real number $m$ such that $|p(z)| \leq m$ for all complex numbers $z$ with $|z| \leq r$. |
The offset polynomial of $0$ is $0$. |
The offset polynomial of a polynomial with a leading coefficient is the product of the offset polynomial of the rest of the polynomial with the offset, plus the offset polynomial of the rest of the polynomial. |
The offset of a constant polynomial is itself. |
The polynomial obtained by shifting the argument of a polynomial $p$ by $h$ is the same as the polynomial obtained by shifting the coefficients of $p$ by $h$. |
If $p$ is a polynomial and $p(x) = c p(x - a)$ for some $c$ and $a$, then $p$ is the zero polynomial. |
The offset polynomial of a polynomial $p$ is zero if and only if $p$ is zero. |
The degree of a polynomial $p$ is the same as the degree of the polynomial $p(x + h)$. |
The polynomial $p$ is zero if and only if its size is zero. |
For any polynomial $p$, there exists a polynomial $q$ such that $p(x) = q(x + a)$ for all $x$. |
If there exists an $x$ such that $P(x)$ is true, and there exists a $z$ such that $P(x)$ implies $x < z$, then there exists a real number $s$ such that $y < s$ if and only if there exists an $x$ such that $P(x)$ is true and $y < x$. |
If $z$ is unimodular, then $z + 1$, $z - 1$, $z + i$, or $z - i$ has modulus less than 1. |
If $b \neq 0$ and $n \neq 0$, then there exists a complex number $z$ such that $|1 + bz^n| < 1$. |
The distance between two complex numbers is bounded by the sum of the distances between their real and imaginary parts. |
If the sequence $(s_n)$ is bounded, then it has a convergent subsequence. |
If $p$ is a polynomial and $z$ is a complex number, then there exists a $\delta > 0$ such that for all $w$ with $|w - z| < \delta$, we have $|p(w) - p(z)| < \epsilon$. |
There exists a complex number $z$ such that for all $w$ with $|w| \leq r$, we have $|p(z)| \leq |p(w)|$. |
If $p$ is a nonzero polynomial, then there exists a real number $r$ such that for all complex numbers $z$ with $|z| \geq r$, we have $|p(z)| \geq d$. |
There exists a complex number $z$ such that for all complex numbers $w$, we have $|p(z)| \leq |p(w)|$. |
If a polynomial is not constant, then its length is at least 2. |
If $p$ is a polynomial, then $x^n p(x)$ is the same as $x^n$ times $p(x)$. |
If $p$ is a non-zero polynomial, then there exists a non-zero constant $a$ and a polynomial $q$ such that $p(z) = z^k a q(z)$ for some $k \geq 0$. |
If a polynomial is not constant, then it can be written as $p(z) = p(0) + z^k p_1(z)$, where $p_1$ is a polynomial of degree $n-k-1$ and $k$ is a positive integer. |
If a polynomial $p$ is not constant, then it has a root. |
If a polynomial $p$ has no constant term, then it has a root. |
If $p$ and $q$ are polynomials such that $p(x) = 0$ implies $q(x) = 0$, then $p$ divides $q^n$ if $p$ has degree $n$. |
If $p$ and $q$ are univariate polynomials, then $p$ divides $q^{\deg(p)}$ or $p = q = 0$. |
A polynomial is constant if and only if its degree is zero. |
The polynomial $0$ is the zero polynomial, the polynomial $c$ is the constant polynomial $c$, and the polynomial $x$ is the linear polynomial $x$. |
The polynomial $0$ is the same as the polynomial $[0]$. |
For any polynomial $p$ and $q$ and any ring element $x$, $p(x) - q(x) = p(x) + (-1)q(x)$. |
If $p(x) = 0$, then $p(x) = 0$. |
If $x = 0$ and $a \neq 0$, then $y = 0$ if and only if $a y - b x = 0$. |
If $p$ divides $q$, then $p$ divides $p \cdot 0 + q$. |
If $p$ is a nonzero polynomial and $q$ is a polynomial with degree less than the degree of $p$, then $p$ divides $q$ if and only if $q = 0$. |
If $p$ divides $p'$, $q$ and $r$ are polynomials such that $aq - p' = r$, then $p$ divides $q$ if and only if $p$ divides $r$. |
The following are equivalent: $\exists x. p(x) = 0 \land 0(x) \neq 0$ $\exists x. 0(x) \neq 0$ $\exists x. c(x) \neq 0$ $\exists x. 0(x) = 0$ $\exists x. c(x) = 0$ |
If $p$ is a non-zero polynomial, then there exists a complex number $x$ such that $p(x) = 0$. |
For a polynomial $p$, the statement $\exists x. p(x) \neq 0$ is equivalent to the statement $p \neq 0$. |
If $p$ is a nonzero polynomial, then there exists a complex number $x$ such that $p(x) = 0$ and $q(x) \neq 0$ if and only if $p$ does not divide $q^n$, where $n$ is the degree of $p$. |
If $q^n$ and $r$ have the same values at all points, then $p$ divides $q^n$ if and only if $p$ divides $r$. |
For any commutative ring $R$ and any $x \in R$, the polynomial $p(x) = c$ is equal to $y$ if and only if $c = y$. |
If $f$ is a holomorphic function on a contractible set $S$ and $f(a) \neq \pm 1$, then there exists a holomorphic function $g$ on $S$ such that $f(z) = \cos(\pi g(z))$ for all $z \in S$. |
For any positive integer $n$, we have $n + \sqrt{n^2 - 1} > 0$. |
For any real number $x \geq 0$, there exists a positive integer $n$ such that $\left|x - \frac{\ln(n + \sqrt{n^2 - 1})}{\pi}\right| < \frac{1}{2}$. |
If $z$ is a complex number of the form $m + \frac{1}{\pi} \ln(n + \sqrt{n^2 - 1})$ or $m - \frac{1}{\pi} \ln(n + \sqrt{n^2 - 1})$ for some integers $m$ and $n$, then $\cos(\pi \cos(\pi z))$ is either $1$ or $-1$. |
If $f$ is a holomorphic function on the unit disk, $f(0)$ is bounded by $r$, $f$ does not take the values $0$ or $1$ on the unit disk, and $z$ is a point in the disk with norm less than $t$, then $f(z)$ is bounded by an exponential function of $t$. |
For every holomorphic function $f$ defined on a neighborhood of the origin, there exists a radius $R$ such that if $f$ is nonzero and nonconstant on the disk of radius $R$, then the derivative of $f$ at the origin is less than $1$ in absolute value. |
If $f$ is a holomorphic function on the complex plane such that $f(z) \neq 0$ and $f(z) \neq 1$ for all $z$, then $f$ is constant. |
If $f$ is a holomorphic function on the complex plane and $f$ does not take on the values $a$ and $b$, then $f$ is constant. |
If $f$ is a holomorphic function on the complex plane that is periodic with period $p \neq 0$, then $f$ has a fixed point. |
If $f$ is a holomorphic function on the complex plane that is not a translation, then there exists a point $x$ such that $f(f(x)) = x$. |
Suppose that for each $i$ and each sequence $r$, there exists a strictly increasing subsequence $k$ of $r$ such that $P(i, k)$ holds. If $P(i, k)$ holds for some strictly increasing subsequence $k$ of $r$, then $P(i, k')$ holds for any strictly increasing subsequence $k'$ of $r$ that is eventually equal to $k$. Then th... |
If $f$ is a sequence of functions from a countable set $S$ to a normed vector space, and if the norm of $f_n(x)$ is bounded by $M$ for all $n$ and $x \in S$, then there exists a subsequence of $f$ that converges for all $x \in S$. |
Suppose $\mathcal{F}$ is a sequence of continuous functions defined on a compact set $S$. If the functions in $\mathcal{F}$ are uniformly bounded and equicontinuous, then there exists a subsequence $\mathcal{F}'$ of $\mathcal{F}$ that converges uniformly to a continuous function $g$ on $S$. |
Suppose $S$ is an open set in the complex plane, and $\mathcal{F}$ is a sequence of holomorphic functions on $S$ such that the range of $\mathcal{F}$ is a bounded set. Then there exists a subsequence $\mathcal{F}'$ of $\mathcal{F}$ and a holomorphic function $g$ on $S$ such that $\mathcal{F}'$ converges uniformly to $g... |
If $g$ is a uniform limit of a sequence of holomorphic functions $\{f_n\}$ on an open connected set $S$, and $g$ is not constant on $S$, then $g$ is not zero on $S$. |
If a sequence of holomorphic functions $\{f_n\}$ converges uniformly to a holomorphic function $g$ on a connected open set $S$, and if each $f_n$ is injective on $S$, then $g$ is injective on $S$. |
Suppose $S$ is an open connected set, $w \in S$, $r > 0$, and $Y \subseteq X$ is a set of holomorphic functions on $S$ such that $h(w) \leq r$ for all $h \in Y$. Then there exists a positive number $B$ and an open set $Z$ containing $w$ such that $h(z) \leq B$ for all $h \in Y$ and $z \in Z$. |
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