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The quotient metric is the largest metric on a quotient set for which the quotient map is non-expansive. It is obtained by taking infima of lengths of chains that may jump freely within equivalence classes.

A non-expansive map between metric spaces is a function $f$ such that $d(fx,fy)\le d(x,y)$.

The complete theory of infinite-dimensional vector spaces over a fixed finite field is aleph-zero-categorical: every countably infinite model has countably infinite dimension and is therefore isomorphic to every other such model.

Compactness gives a countable model elementarily equivalent to $(\mathbb{Z},+,0)$ that contains a nonzero element divisible by every positive standard integer. It is not isomorphic to $\mathbb{Z}$, so $\mathrm{Th}(\mathbb{Z},+,0)$ is not aleph-zero-categorical.

A finite limit is a categorical limit whose indexing category has finitely many objects and morphisms.

Absolute thermodynamic temperature can be defined so that a reversible engine exchanging heats $Q_h$ and $Q_c$ with reservoirs at $T_h$ and $T_c$ satisfiesA choice of one reference temperature fixes the scale.

The continuum hypothesis asserts that every infinite subset of the real numbers is either countable or equinumerous with the real numbers, equivalently $2^{{\aleph }_0}={\aleph }_1$.

For infinite cardinals $\kappa ,\lambda$, Hausdorff's formula is

For infinite cardinals, choice gives $\kappa +\kappa =\kappa \cdot \kappa =\kappa$. Exponentiation satisfies $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \mu }$, while a countable sum of cardinals has size the maximum of ${\aleph }_0$ and their supremum.

In ZF, let $K,L$ be nonempty. If $K\times L$ injects into $K\bigsqcup L$, then there is either an injection or a surjection from $K$ to $L$.

Indeed, extend the inverse of the given injection to a surjection $p:K\bigsqcup L\to K\times L$ by sending points outside its range to one fixed pair $(k_{\ast },l_{\ast })$. If the second coordinate of $p$ restricted to the $K$-summand covers $L$, it is the required surjection. Otherwise choose $l_0$ that it misses. Every $(k,l_0)$ then has its preimage in the $L$-summand. Except possibly for the default pair, that preimage is unique, so these preimages inject $K$ into $L$. In the exceptional case they inject $K\setminus \{k_{\ast }\}$ into $L$; either their image is all of $L$, yielding a surjection from $K$, or an omitted point extends the map to an injection from $K$.

For an oriented closed connected $d$-manifold, the fundamental class is the unique generator of $H_d(M;\mathbb{Z})$ that restricts to the orientation generator in every local homology group.

The leading term ${\omega }^{\delta }n$ is the highest-exponent term in Cantor normal form; the remaining tail is strictly below ${\omega }^{\delta }$.

The Belinfante-Rosenfeld construction adds the divergence of a spin-current superpotential to the canonical stress-energy tensor, producing a symmetric conserved tensor with the same integrated four-momentum under suitable boundary conditions.

For a Poisson mean $\mu$, the natural parameter is $\theta =\log \mu$, so the canonical link is the logarithmic link $g(\mu )=\log \mu$.

If a nondecreasing oscillation quantity satisfiesfor $0<\rho \le R$, with $\gamma >\beta$, then iteration gives $\Phi (\rho )\le C(\rho /R{)}^{\beta }\Phi (R)+CB{\rho }^{\beta }$.

An operator $U$ on a Hilbert space is a partial isometry when it is isometric on $(\ker U{)}^{\perp }$. Equivalently, $U^{\ast }U$ is the orthogonal projection onto this initial space.

An element $a$ of a C-star algebra is positive when $a=b^{\ast }b$ for some $b$, equivalently when $a=a^{\ast }$ and $\sigma (a)\subseteq [0,\infty )$. Continuous functional calculus gives it a unique positive square root $a^{1/2}$.

If $(U,V)$ is $C$-relevant to $(X,Y)$ and $U_1,U_2,V_1,V_2$ are mutually independent copies, then $(U_1+U_2,V_1+V_2)$ is $2C$-relevant to $(X,Y)$. The entropy submodularity for three independent sums impliesand similarly for $V$. The Entropic Ruzsa triangle inequality boundswhich proves the claim after addition.

For distinct accessible subset states, choose a state in their symmetric difference. A word taking it to the unique final state accepts from one subset; uniqueness of the starting state prevents acceptance from the other. Thus the accessible part of the subset automaton is irreducible.

Let $W$ be continuous with $W_0=0$. Ifis a martingale for every real $c$, then conditional expectations solveThus $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and distributed as $N(0,t-s)$, so $W$ is Brownian motion.