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If distinct points $x,y$ of a dynamical system on a compact metric space satisfy $d(T^{n}x,T^{n}y)\to 0$ in one compatible metric, no compatible metric can make $T$ an isometry. On a compact space, all compatible metrics give the same asymptotic-pair property by uniform continuity, whereas an isometry preserves the strictly positive distance between distinct points. In a full shift, a constant sequence and a sequence differing at just one coordinate converge to each other under forward shifts, proving this obstruction directly in the product topology.

A Borel probability measure is a Borel measure with total mass one. On a compact metric space it is determined by its integrals against continuous functions. On that compact metric space, the family of Borel probability measures is compact for weak convergence of probability measures.

A subset $A$ of a metric space is bounded if it lies in some finite-radius ball: there are $x$ and $R<\infty$ such that $d(x,a)\le R$ for every $a\in A$.

The category ${\mathbf{Met}}_b$ is the full subcategory on bounded metric spaces. It has finite products: the terminal space is a singleton and

The category $\mathbf{Met}$ has metric spaces as objects and maps $f$ satisfying $d(fx,fy)\le d(x,y)$ as morphisms. Its binary product uses the maximum metric.

In a metric space $(M,d)$, the closed ball centred at $x$ of radius $r\ge 0$ is $\{y\in M:d(x,y)\le r\}$. It is a closed set, since the distance to $x$ is a continuous function. Its boundary need not be the whole corresponding distance sphere in an arbitrary metric space.

An isometric group action on a metric space is cobounded if some orbit is within a fixed distance $R$ of every point. Equivalently, translates of one bounded ball cover the space. Coboundedness does not require finite stabilizers, a proper space, or a finitely generated group. Every action on a point is cobounded.

A compact metric space is a metric space that is compact. Equivalently, it is complete and totally bounded.

A metric is compatible with a given topology when its open balls generate exactly that topology. Compatible metrics on a compact topological space are uniformly equivalent: the identity maps between the resulting compact metric spaces are uniformly continuous. Consequently they define the same proximality and the same asymptotic-pair relation for a fixed continuous map.

For a metric space with a probability measure $\mu$, the concentration function is $\alpha (\epsilon )={\sup }_{\mu (A)\ge 1/2}(1-\mu (A_{\epsilon }))$, where $A_{\epsilon }$ is the metric neighbourhood of radius $\epsilon$. A sequence exhibits concentration of measure if these functions tend to zero for every fixed positive radius. The metric scale is part of this definition.

A metric space is complete if and only if every contraction mapping of every nonempty closed subset into itself has a fixed point. For the converse, a nonconvergent Cauchy sequence contains a rapidly approaching subsequence whose point set is closed and whose shift is a contraction without a fixed point. Requiring the property only on the whole space is a different, weaker condition.

For a point $x$ and a nonempty closed set $A$ in a metric space, the distanceis a continuous function of $x$. It vanishes exactly on $A$.

For observations or orbit points $x_0,\dots ,x_{N-1}$, the empirical measure is ${\nu }_N=N^{-1}{\sum }_{j=0}^{N-1}{\delta }_{x_j}$. Integrating a test function gives its sample average. For orbit points $x_j=T^{j}x$, the difference between $T_{\ast }{\nu }_N$ and ${\nu }_N$ telescopes to $({\delta }_{T^{N}x}-{\delta }_x)/N$, so on a compact metric space with continuous $T$, every weak limit is invariant.

On a compact metric space, a conservative Feller semigroup is a strongly continuous semigroup of positive unital operators on $C(X)$. The Riesz-Markov-Kakutani representation theorem associates a Markov kernel to each operator. On noncompact spaces the standard definition uses $C_0(X)$; conservativity then refers to the corresponding kernels having mass one, even when $1\notin C_0(X)$.

An $\epsilon$-net of a metric space is a set $G$ such that every point lies within distance $\epsilon$ of a point of $G$. A Cartesian grid of coordinate spacing $\eta$ is a finite $\sqrt{d}\eta$-net of a bounded cube in the Euclidean norm. A Lipschitz bound transfers control on the grid to control over the whole cube.

For four points $x_1,x_2,y_1,y_2$ in a metric space,Both steps follow from the triangle inequality. The first also holds with an absolute value around the difference, by the reverse triangle inequality. This transports a difference of distances measured from one point into a sum measured from another.

A Frechet embedding uses distance functions as coordinates, typicallyfor selected subsets $A_i$. Every coordinate is one-Lipschitz, and suitable subsets provide lower bounds on pairwise image distances.

A subspace of a complete metric space admits a compatible complete metric if and only if it is a G-delta set. For the reverse direction, adjoining the coordinates $1/d(x,X\setminus U_n)$ to the ambient metric prevents Cauchy sequences from approaching any excluded closed boundary. For the forward direction, small-diameter relative open covers in a compatible complete metric produce the ambient open sets.