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A metric space in which every two points are joined by a metric geodesic segment. Equivalently, they admit a continuous path whose metric path length equals their distance. In this general setting no Riemannian metric or covariant derivative is required.

For compact metric spaces $X$ and $Y$, the Gromov-Hausdorff distance iswhere the infimum runs over all metric spaces $Z$ and all isometric embeddings $\phi :X\to Z$ and $\psi :Y\to Z$. It measures how closely the two spaces can be placed inside one ambient metric space.

For a locally finite discrete metric space $X$ with basepoint $x_0$, its growth function isChanging the basepoint changes the argument by at most an additive constant. In a vertex-transitive metric space, including every Cayley graph, the growth function is exactly independent of the basepoint.

An inclusion-maximal separated set is one to which no further point of the ambient metric space can be added while keeping the same separation. It need not have the largest possible cardinality. In a finite space it can be obtained by repeatedly adding an admissible point.

An isometric embedding $f:X\to Y$ between metric spaces is an injective function satisfying $d_Y(f(x),f(x^{\prime }))=d_X(x,x^{\prime })$ for all $x,x^{\prime }\in X$.

An isometry between metric spaces preserves every distance. A Riemannian isometry preserves the Riemannian metric and therefore distances, angles, geodesics, and intrinsic curvature.

The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

For a compact metric space with a continuous map $T$, suppose $y$ is a minimal point and $x,y$ are proximal. Every neighborhood $U$ of $y$ admits arbitrarily large $n$ with $T^{n}x,T^{n}y\in U$. First shrink $U$ to an open neighborhood, then choose an open $V$ with $y\in V$ and $\overline{V}\subseteq U$. Minimality and compactness give a finite cover of the orbit closure of $y$ by $T^{-j}V$, $0\le j\le J$. Uniform continuity of these finitely many iterates turns a sufficiently close proximal encounter into a common visit to $U$ after at most $J$ more steps. If the proximal encounters occur only at bounded times, a zero distance at some time makes the two future orbits coincide, and minimality then gives arbitrarily late common visits. Thus injectivity is not needed for this general lemma.

For a Feller semigroup on a compact metric space, time-averaged transition laws have weakly convergent subsequences by compactness of probability measures on a compact metric space. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.

If an inclusion-maximal separated set has separation at least $r$, every point of the ambient metric space is at distance strictly less than $r$ from a selected point. Otherwise that point could be added. Thus upper bounds on the sizes or measures of radius-$r$ balls give lower bounds on the size of the separated set.

A metric on a set $X$ is a function $d:X\times X\to [0,\infty )$ that is symmetric, satisfies the triangle inequality, and obeys $d(x,y)=0$ exactly when $x=y$.

For a metric space or pseudometric space $(T,d)$, this is the minimum number of balls of radius $\epsilon >0$, with centers in $T$, needed to cover $T$. The displayed grid bound uses closed balls. For open balls, replacing the radius by a fixed constant factor gives the same entropy estimates. For $T=[0,1]$ and $d(s,t)=c|s-t{|}^H$, a uniform grid gives $N(\epsilon ,T,d)\le 1+\lceil (c/\epsilon {)}^{1/H}\rceil$. These polynomial growth bounds make the Dudley entropy integral finite.

A unit-speed metric geodesic in a metric space is an isometric embedding $\gamma :I\to X$ of a real interval. Every restricted segment realizes the distance between its endpoints. A one-point interval permits a constant segment. In a unit-edge graph, shortest edge paths parametrized by metric path length are metric geodesics; no smooth structure is required. This is stronger than the differential definition of a geodesic in a Riemannian manifold, which need only minimize locally: on a unit circle, an arc of angle $3\pi /2$ is a Riemannian geodesic but its endpoint distance is only $\pi /2$.

An $\epsilon$-net of a metric space $(T,d)$ is a subset $U\subseteq T$ such that every $x\in T$ has $d(x,u)\le \epsilon$ for some $u\in U$. Its least possible finite size is the metric covering number with closed balls. A maximal set with pairwise distances greater than $\epsilon$ is an $\epsilon$-net.

For a continuous path $\gamma :[a,b]\to X$ in a metric space, take the supremum over finite partitions $a=t_0<\cdots <t_n=b$ of ${\sum }_{i=1}^{n}d(\gamma (t_i),\gamma (t_{i-1}))$. This definition allows $+\infty$ and needs no derivative. A metric geodesic parametrized on $[a,b]$ has length $b-a$. In a Riemannian manifold, it agrees with the usual arc length integral for a smooth curve.

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

In a metric space, the open ball of radius $r$ about $x$ is $\{y:d(x,y)<r\}$.

A topological space is paracompact when every open cover has a locally finite open refinement. Every metric space and every second-countable topological manifold is paracompact.