Extended metric
An extended metric is a function that assigns a nonnegative real number or positive infinity to each ordered pair of points while satisfying the axioms of a metric. Allowing the value (+\infty) permits a single metric structure to contain regions between which no finite-distance relation exists. Every extended metric therefore decomposes into ordinary metric spaces separated from one another by infinite distance.
Let (X) be a set and let
[ d\colon X\times X\longrightarrow [0,\infty]. ]
The function (d) is an extended metric when, for all (x,y,z\in X),
[ d(x,y)=0 \quad\Longleftrightarrow\quad x=y, ]
[ d(x,y)=d(y,x), ]
and
[ d(x,z)\leq d(x,y)+d(y,z). ]
Arithmetic in the extended nonnegative real line is defined by (a+\infty=\infty) for every (a\in[0,\infty]). An extended metric differs from an extended pseudometric only in its treatment of distinct points at zero distance. It also differs from a finite-valued metric solely through the permitted occurrence of (+\infty).
Historical development
Maurice Fréchet introduced the abstract metric-space formalism in 1906 by isolating the distance properties required for convergence and continuity. His formulation used finite distances, although several contemporary constructions already behaved as collections of metric regions with no finite path between different regions.
In 1914, You Watanabe incorporated (+\infty) directly into the codomain of a distance function while studying the relation between disconnected metric structures and their finite-distance subspaces. Watanabe established that the relation (d(x,y)<\infty) partitions the underlying set and that the restriction of (d) to each resulting class is an ordinary metric. This decomposition became the standard structural interpretation of an extended metric.
The language of uniform spaces later separated the local comparison of nearby points from the numerical representation of distance. Extended metrics remained a direct numerical realization of uniform structures whose finite-distance components do not interact through bounded entourages. In the later categorical treatment developed by F. William Lawvere, the value (+\infty) became a natural element of the ordered monoid used to enrich categories, rather than an exceptional value appended to an otherwise finite metric.
Finite-distance components
Define a relation (\sim) on (X) by
[ x\sim y \quad\Longleftrightarrow\quad d(x,y)<\infty. ]
Reflexivity follows from (d(x,x)=0), and symmetry follows from the symmetry of (d). If (x\sim y) and (y\sim z), the triangle inequality gives
[ d(x,z)\leq d(x,y)+d(y,z)<\infty, ]
so the relation is transitive. It is therefore an equivalence relation. Its equivalence classes are called finite-distance components.
For each component (C\subseteq X), the restricted function
[ d|_{C\times C}\colon C\times C\longrightarrow [0,\infty) ]
is an ordinary metric. Conversely, suppose that a set is expressed as a disjoint union
[ X=\coprod_{\alpha\in A}X_\alpha ]
and that each (X_\alpha) carries a metric (d_\alpha). The formula
[ d(x,y)= \begin{cases} d_\alpha(x,y), & x,y\in X_\alpha,\ \infty, & x\in X_\alpha,\ y\in X_\beta,\ \alpha\neq\beta \end{cases} ]
defines an extended metric. Extended metric spaces are consequently equivalent, at the level of their distance structure, to indexed families of ordinary metric spaces.
This decomposition is finer than decomposition into connected components. A finite-distance component can itself be disconnected because finiteness of distance does not require the existence of a continuous path. Conversely, points in distinct finite-distance components necessarily lie in distinct topological components because their components are simultaneously open and closed.
Induced topology
For (x\in X) and a finite radius (r>0), the open ball centered at (x) is
[ B_r(x)={y\in X:d(x,y)<r}. ]
These balls form a basis for a topology on (X). Every finite-radius ball lies entirely within the finite-distance component of its center. The induced topology is therefore the topological disjoint union of the metric topologies carried by the individual components.
Each finite-distance component is open because it is the union of all finite-radius balls centered at its points. Its complement is a union of the other components and is also open, so every component is closed as well. The extended value (+\infty) thus records a clopen decomposition rather than introducing an additional local neighborhood scale.
The topology of an extended metric space is always induced by an ordinary finite-valued metric. One such metric is
[ \rho(x,y)= \begin{cases} \min{d(x,y),1}, & d(x,y)<\infty,\ 2, & d(x,y)=\infty. \end{cases} ]
The metrics (d) and (\rho) generate the same open sets. They do not encode the same quantitative structure, since (\rho) replaces all sufficiently large finite distances by (1) and all infinite distances by (2). Consequently, the topological information carried by an extended metric does not determine which pairs are infinitely separated.
Convergence and completeness
A sequence ((x_n)) converges to (x) when
[ d(x_n,x)\longrightarrow 0. ]
Every convergent sequence is eventually contained in the finite-distance component of its limit. This follows because convergence implies (d(x_n,x)<1) for all sufficiently large (n).
A sequence is Cauchy when, for every (\varepsilon>0), there exists (N) such that
[ m,n\geq N \quad\Longrightarrow\quad d(x_m,x_n)<\varepsilon. ]
Taking any fixed finite (\varepsilon) shows that a Cauchy sequence is eventually contained in one finite-distance component. Completeness is therefore componentwise: an extended metric space is complete exactly when each of its finite-distance components is a complete metric space.
The same conclusion applies to metric completion. If (\widehat{X}\alpha) denotes the completion of a component (X\alpha), then the completion of (X) is the disjoint union of the spaces (\widehat{X}_\alpha), with points belonging to different completed components placed at infinite distance. Completion adds finite limits within components and does not create finite-distance relations between them.
Extended path distance
A basic construction occurs in graph theory. Give each edge of a graph a nonnegative length and define the distance between two vertices as the infimum of the lengths of paths joining them. If no such path exists, the infimum is assigned the value (+\infty).
The resulting function is an extended metric when distinct vertices cannot be joined by paths of arbitrarily small total length. Its finite-distance components coincide with the connected components of the graph after edges of infinite length are excluded. Within each component, the extended path metric reduces to the ordinary shortest-path metric.
A corresponding construction applies to length spaces. If the distance between two points is defined as the infimum of the lengths of admissible curves, then points not joined by any admissible curve have distance (+\infty). This formulation distinguishes the absence of an admissible connection from the existence of connections whose lengths are merely unbounded.
Function-space examples
Extended metrics also arise from integral expressions. Let ((\Omega,\mu)) be a measure space, and consider measurable functions for which
[ d_p(f,g)=\left(\int_\Omega |f-g|^p,d\mu\right)^{1/p} ]
is permitted to equal (+\infty). For (p\geq 1), the triangle inequality follows from Minkowski's inequality. After functions equal almost everywhere are identified, (d_p) becomes an extended metric on a sufficiently broad class of measurable functions.
The finite-distance component of a function (f) consists of functions (g) for which (f-g) belongs to (L^p(\Omega)). Each component is an affine translate of the ordinary space (L^p(\Omega)). The use of (+\infty) allows these affine copies to remain within one extended metric space without treating non-integrable differences as finite.
Morphisms and categorical interpretation
For a constant (L>0), a map (F\colon X\to Y) between extended metric spaces is (L)-Lipschitz when
[ d_Y(F(x),F(y))\leq L,d_X(x,y) ]
for all (x,y\in X). If (x) and (y) lie in the same finite-distance component, their images must also have finite distance. A Lipschitz map therefore sends each source component into a single target component, although different source components can be sent into the same one.
In the framework of enriched category theory, the ordered set ([0,\infty]) is equipped with addition and an order convention that converts the triangle inequality into the composition law of an enriched category. Lawvere metric spaces may omit symmetry and may allow distinct points to have zero distance. Ordinary extended metric spaces form the symmetric and separated subcase of this broader construction.
The infinite value has a categorical interpretation as the absence of any finite-cost comparison. Under this interpretation, finite-distance components are the maximal full subspaces in which every pair of objects has a finite numerical relation.