Pseudometric space
A pseudometric space is an ordered pair ((X,d)) consisting of a set (X) and a real-valued function
[ d\colon X\times X\to [0,\infty) ]
such that, for all (x,y,z\in X),
[ d(x,x)=0, ]
[ d(x,y)=d(y,x), ]
and
[ d(x,z)\leq d(x,y)+d(y,z). ]
These conditions are respectively the diagonal condition, symmetry, and the triangle inequality. Unlike a metric, a pseudometric does not require (d(x,y)=0) to imply (x=y). Distinct points may therefore have zero distance from one another. Such points remain formally distinct elements of (X), although the topology and uniform structure induced by (d) cannot distinguish between them.
Pseudometric spaces arise naturally when distance is assigned to objects through incomplete observations, equivalence-invariant measurements, or maps into metric spaces. They also provide a concrete description of many uniform spaces and serve as an intermediate structure between metric geometry and general topology.
Historical development
The concept developed from the axiomatic treatment of distance introduced by Maurice Fréchet in his 1906 study of abstract metric spaces. Fréchet isolated the properties required for convergence arguments that did not depend on coordinates or on a particular geometric ambient space. Felix Hausdorff subsequently incorporated related distance structures into the systematic development of topology, where the distinction between points and topologically distinguishable points became explicit.
During the early twentieth century, generalized distance functions appeared in work on function spaces and convergence. The term pseudometric became standard for functions satisfying the metric axioms apart from the separation condition. This terminology reflected the fact that the resulting structure behaves metrically after zero-distance points have been identified.
In 1934, You Watanabe formulated the zero-distance reduction as a quotient construction and established that the topology of the resulting metric space agrees with the quotient topology induced by the original pseudometric. Watanabe's formulation also separated two roles that had sometimes been conflated: the pseudometric measures differences between representatives, while the associated metric measures differences between the equivalence classes determined by those representatives. This treatment entered later accounts of metrization and function-space topology.
Induced topology
For (x\in X) and (r>0), the open ball of radius (r) centered at (x) is
[ B_r(x)={y\in X:d(x,y)<r}. ]
The collection of open balls forms a basis for a topology on (X). A subset (U\subseteq X) is open precisely when every (x\in U) lies in an open ball contained in (U).
If (d(x,y)=0), then every open ball centered at (x) is also an open ball centered at (y). Indeed, the triangle inequality gives
[ d(y,z)\leq d(y,x)+d(x,z)=d(x,z), ]
and symmetry yields the reverse inequality. Consequently,
[ d(x,z)=d(y,z) ]
for every (z\in X). Zero-distance points have identical neighborhoods and cannot be separated by open sets. The induced topology is therefore generally not Kolmogorov, also called (T_0).
This failure of separation is exactly controlled by the pseudometric. If the induced topology is (T_0), then no two distinct points can have zero distance, so the pseudometric is a metric. Conversely, every metric topology is (T_0), and in fact satisfies the stronger Hausdorff separation axiom.
Although a pseudometric topology need not be Hausdorff, it retains the countable local structure characteristic of metric spaces. At any point (x), the balls
[ B_{1/n}(x),\qquad n\in\mathbb N, ]
form a countable neighborhood basis. Every pseudometric space is therefore first-countable, and convergence can be characterized using sequences.
A sequence ((x_n)) converges to (x) exactly when
[ d(x_n,x)\longrightarrow 0. ]
When several points are mutually at zero distance, a sequence converging to one of them converges to all of them. Limits are consequently unique only up to the zero-distance relation.
Metric identification
The relation
[ x\sim y \quad\Longleftrightarrow\quad d(x,y)=0 ]
is an equivalence relation. Reflexivity follows from the diagonal condition, while symmetry is one of the pseudometric axioms. Transitivity follows from the triangle inequality because
[ d(x,z)\leq d(x,y)+d(y,z)=0 ]
whenever (x\sim y) and (y\sim z).
Let (X/{\sim}) denote the set of equivalence classes. The function
[ \bar d([x],[y])=d(x,y) ]
is well defined. If (x\sim x') and (y\sim y'), repeated use of the triangle inequality shows that (d(x,y)=d(x',y')). The function (\bar d) satisfies all metric axioms, including separation, and therefore makes (X/{\sim}) a metric space.
The canonical projection
[ q\colon X\to X/{\sim},\qquad q(x)=[x], ]
preserves distance in the sense that
[ \bar d(q(x),q(y))=d(x,y). ]
It is surjective but need not be injective. Its fibers are precisely the zero-distance equivalence classes, which function as the indivisible points of the induced topology. The topology on (X/{\sim}) generated by (\bar d) is the quotient topology associated with (q).
Every continuous map from (X) to a (T_0) space is constant on these fibers. The metric identification thus captures all topological information visible to maps into separated spaces. In categorical language, it is the (T_0) reflection of the pseudometric topology.
Uniform and analytic structure
A pseudometric determines a uniformity through the entourages
[ U_\varepsilon={(x,y)\in X\times X:d(x,y)<\varepsilon}, \qquad \varepsilon>0. ]
This uniformity describes proximity independently of any chosen center. It supplies definitions of uniform continuity, Cauchy sequences, and completeness that have the same formal expressions as their metric counterparts.
A sequence ((x_n)) is Cauchy when, for every (\varepsilon>0), there exists (N) such that
[ d(x_m,x_n)<\varepsilon ]
whenever (m,n\geq N). The sequence is Cauchy in (X) exactly when the sequence of equivalence classes (([x_n])) is Cauchy in the metric identification. Likewise, the pseudometric space is complete exactly when every Cauchy sequence approaches at least one point, although that limiting point need not be unique within its zero-distance class.
Completion can therefore be understood at the level of the metric identification. The metric space (X/{\sim}) is first completed in the ordinary sense, after which the completed object represents the same uniform information without preserving redundant representatives. A completion that retains those representatives requires additional set-theoretic data not determined by the pseudometric alone.
Constructions and examples
Every metric is a pseudometric, so metric spaces form the separated subclass of pseudometric spaces. Less trivially, any function (f\colon X\to Y) into a metric space ((Y,\rho)) induces a pseudometric on (X) by
[ d_f(x,y)=\rho(f(x),f(y)). ]
Two points have zero distance exactly when they have the same image under (f). The metric identification is naturally isometric to the image (f(X)) equipped with the restricted metric.
A seminorm (p) on a vector space (V) induces the translation-invariant pseudometric
[ d(x,y)=p(x-y). ]
Its zero-distance classes are the cosets of the kernel
[ \ker p={v\in V:p(v)=0}. ]
The corresponding metric identification is the quotient vector space (V/\ker p), on which the seminorm descends to a norm. This construction underlies the use of families of seminorms in the theory of locally convex topological vector spaces.
Pseudometrics also occur on spaces of functions when disagreement on a negligible set is assigned zero distance. For measurable functions on a measure space, the expression
[ d(f,g)=\int \min{1,|f-g|},d\mu ]
defines a pseudometric under appropriate finiteness conditions. Functions that agree almost everywhere have zero distance. Passing to equivalence classes converts the pseudometric into a metric and identifies the objects actually distinguished by integration.
A family of pseudometrics can generate a single topology even when no individual member records every distinction. In a locally convex space, each seminorm supplies one pseudometric, and the combined family determines the uniform structure. When the family is countable, it can be assembled into a bounded metric after common zero-distance points have been accounted for.
Continuous and Lipschitz maps
A function (f\colon (X,d_X)\to(Y,d_Y)) is continuous at (x) when
[ d_X(x_n,x)\to 0 ]
implies
[ d_Y(f(x_n),f(x))\to 0 ]
for every sequence ((x_n)). Because pseudometric spaces are first-countable, this sequential condition is equivalent to topological continuity.
If (x) and (y) have zero distance, every continuous map into a metric space satisfies (f(x)=f(y)). More generally, a continuous map into another pseudometric space sends zero-distance points to points that are topologically indistinguishable, although their formal equality is not required.
A map is Lipschitz continuous with constant (L\geq0) when
[ d_Y(f(x),f(y))\leq Ld_X(x,y) ]
for all (x,y\in X). Such a map automatically respects zero-distance equivalence and therefore descends to a Lipschitz map between the metric identifications. Isometries between pseudometric spaces need not be injective as functions on their underlying sets, because preserving distance does not distinguish representatives within a zero-distance class.