In mathematics, a completely metrizable space (metrically topologically complete space) is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

(''X'', ''T'') for which there exists at least one

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

''d'' on ''X'' such that (''X'', ''d'') is a

complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...

and ''d'' induces the topology ''T''. The term topologically complete space is employed by some authors as a synonym for ''completely metrizable space'', but sometimes also used for other classes of topological spaces, like

completely uniformizable space In mathematics, a topological space (''X'', ''T'') is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology ''T''. Some authors additionally require ''X'' to be Hausdorf ...

s or Čech-complete spaces.

Difference between ''complete metric space'' and ''completely metrizable space''

The difference between ''completely metrizable space'' and ''complete metric space'' is in the words ''there exists at least one metric'' in the definition of completely metrizable space, which is not the same as ''there is given a metric'' (the latter would yield the definition of complete metric space). Once we make the choice of the metric on a completely metrizable space (out of all the complete metrics compatible with the topology), we get a complete metric space. In other words, the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

of completely metrizable spaces is a subcategory of that of topological spaces, while the category of complete metric spaces is not (instead, it is a subcategory of the category of metric spaces). Complete metrizability is a topological property while completeness is a property of the metric.

Examples

* The space , the open unit interval, is not a complete metric space with its usual metric inherited from , but it is completely metrizable since it is homeomorphic to . * The space of

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

with the subspace topology inherited from is metrizable but not completely metrizable.

Properties

* A topological space ''X'' is completely metrizable if and only if ''X'' is

metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...

and a G_δ in its

Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...

β''X''. * A subspace of a completely metrizable space is completely metrizable if and only if it is in . * A countable product of nonempty metrizable spaces is completely metrizable in the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-s ...

if and only if each factor is completely metrizable. Hence, a product of nonempty metrizable spaces is completely metrizable if and only if at most countably many factors have more than one point and each factor is completely metrizable. * For every metrizable space there exists a completely metrizable space containing it as a dense subspace, since every metric space has a completion. In general, there are many such completely metrizable spaces, since completions of a topological space with respect to different metrics compatible with its topology can give topologically different completions.

Completely metrizable abelian topological groups

When talking about spaces with more structure than just topology, like

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

s, the natural meaning of the words “completely metrizable” would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology. For abelian topological groups and

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

s, “compatible with the extra structure” might mean that the metric is invariant under translations. However, no confusion can arise when talking about an abelian topological group or a topological vector space being completely metrizable: it can be proven that every abelian topological group (and thus also every topological vector space) that is completely metrizable as a topological space (i. e., admits a complete metric that induces its topology) also admits an invariant complete metric that induces its topology. This implies e. g. that every completely metrizable topological vector space is complete. Indeed, a topological vector space is called complete iff its uniformity (induced by its topology and addition operation) is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.

Notes

References