Metrizable space

A space whose topology is generated by some metric via the rule: a point belongs to the closure of a set if and only if it lies at zero distance from the set. If such a metric exists, then it is not unique, unless the space is empty or consists of one point only. In particular, the topology of each metrizable space is generated by a bounded metric. In a metrizable space strong separation axioms (cf. Separation axiom) are satisfied: it is normal and even collectionwise normal. Every metrizable space is paracompact. All metrizable spaces satisfy the first axiom of countability. But none of the named conditions, nor any collection of them, is sufficient for a space to be metrizable. A sufficient condition for metrizability was found by P.S. Urysohn (1923): Every normal space (and even every regular space, A.N. Tikhonov, 1925) with a countable base is metrizable. The first general criterion for metrizability of a space was proposed in 1923 by P.S. Aleksandrov and Urysohn (see [1]). On its basis two subsequent, more precise, criteria for metrizability were developed: 1) a space is metrizable if and only if it is collectionwise normal and has a countable refining set of open coverings; 2) a space is metrizable if and only if has a countable fundamental set of open coverings and satisfies the $ T _ {1} $ separation axiom (the Stone–Arkhangel'skii criterion). Here a set $ \xi $ of open coverings of a space $ X $ is called fundamental if for each point $ x \in X $ and each of its neighbourhoods $ O _ {x} $ there is a covering $ \gamma \in \xi $ and a neighbourhood $ O _ {1x} $ of $ x $ such that every element of $ \gamma $ intersecting $ O _ {1x} $ is contained in $ O _ {x} $. These criteria are connected with the property of unrestricted divisibility and with the following fundamental property of full normality of metrizable spaces. Every open covering $ \gamma $ of a metrizable space $ X $ can be refined to an open covering $ \gamma ^ \prime $ such that for any $ x \in X $ there is an $ U \in \gamma $ for which $ \cup \{ {W \in \gamma ^ \prime } : {x \in W } \} \subset U $.

Based on another important idea — local finiteness — there is an important general criterion for metrizability. The Nagata–Smirnov criterion: A space $ X $ is metrizable if and only if it is regular and has a base decomposing into a countable set of locally finite families (cf. Locally finite family) of sets. Bing's criterion is similar, but instead of locally finite uses discrete families of sets (cf. Discrete family of sets). Convenient versions of the above metrizability criteria are related to the notions of a uniform base and a regular base. A base $ {\mathcal B} $ of a space $ X $ is called regular (uniform) if for every point $ x \in X $ and any of its neighbourhoods $ O _ {x} $ there is a neighbourhood $ O _ {1x} $ of this point such that the number of elements of the base $ {\mathcal B} $ simultaneously intersecting $ O _ {1x} $ and the complement of $ O _ {x} $ is finite (respectively, if the set $ \{ {U \in {\mathcal B} } : {U \ni x, U \subset O _ {x} } \} $ is finite). A space $ X $ is metrizable if and only if it is collectionwise normal and has a uniform base. Finally, for a $ T _ {1} $- space to be metrizable it is necessary and sufficient that it has a regular base. Regular bases are convenient in that they reveal the mechanism of paracompactness of arbitrary metrizable spaces: In order to inscribe a locally finite open covering inside any open covering $ \gamma $ of a space $ X $ with a regular base $ {\mathcal B} $, it is sufficient to take the collection of all maximal elements of the family

$$ {\mathcal B} _ \gamma = \ \{ {U \in {\mathcal B} } : {\textrm{ there is a } \ W \in \gamma \textrm{ such that } U \subset W } \} . $$

Metrizability criteria become very simple in a number of special classes of spaces. Thus, for a compactum $ X $ to be metrizable, any of the following four conditions is necessary and sufficient: a) $ X $ has a countable base; b) $ X $ has a point-countable base; c) there is a countable network (cf. Net (of sets in a topological space); Network) in $ X $; or d) the diagonal in $ X \times X $ is a $ G _ \delta $- set. For the metrizability of the space of a topological group it is necessary and sufficient that the group satisfies the first axiom of countability; moreover, the space is then metrizable by an invariant metric (for example, with respect to left multiplication).

A characteristic property of a metrizable space is the coincidence of a number of cardinal characteristics. In particular, in a metrizable space the Suslin number, the Lindelöf number, the density, the character, the spread, and the weight all coincide. The non-coincidence of these numbers is an indication of the non-metrizability of the corresponding space.

Not every metrizable space is metrizable by a complete metric: an example is the space of rational numbers. A space is metrizable by a complete metric if and only if it is metrizable and is a set of type $ G _ \delta $ in some compact space containing it. An important topological property of a space metrizable by a complete metric is the Baire property: The intersection of any countable family of everywhere-dense open sets is everywhere dense.

Very close to metrizable spaces in their properties are the so-called Moore spaces, i.e. completely-regular spaces having a countable refining family of open coverings, and lattice spaces.

A broad range of generalizations of the idea of a metrizable space is obtained if the metric axioms are varied, weakening them in some way or other, and by considering the topologies generated by such $ v $- metrics. In this way symmetrizable spaces are obtained by abandoning the triangle axiom. Moore spaces fit into this scheme. Another important generalization of the idea of metrizability is related to the discussion of "metrics" with values in semi-fields and other algebraic structures of a general nature.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2]	R. Engelking, "General topology" , Heldermann (1989)
[3]	M.Ya. Antonovskii, V.G. Boltyanskii, T.A. Sarymsakov, "Metric spaces over semi-fields" , Tashkent (1961) (In Russian)

Comments

The topology of a metrizable space is described here in terms of the closure operation. It is somewhat more common to use the open balls: If for $ x \in X $ and $ \epsilon > 0 $, $ B ( x , \epsilon ) $ denotes the set of points at distance less than $ \epsilon $ from $ x $( the $ \epsilon $- ball around $ x $), then one calls a set $ U $ open if and only if for every $ x \in U $ there is an $ \epsilon > 0 $ such that $ B ( x , \epsilon ) \subseteq U $.

Metrizability criterion 1) is due to R.H. Bing [a1]. A countable refining family of open coverings is also called a development, see (the comments to) Moore space for the definition and more information.

A fundamental collection is sometimes called locally starring. A fully-normal space is also called a star-normal space (every open covering admits a star refinement), especially in the (translated) Russian literature. A regular topological space is fully normal if and only if it is paracompact.

Unrestricted divisibility is a property defined in [a4]; it is analogous to full normality, which is defined above, but not apparently equivalent.

For a survey (and, in part, a survey of surveys) of generalized metrizable spaces, see [a3]. For metrizable spaces, see [a5], especially pages 244-360.

References

[a1]	R.H. Bing, "Metrization of topological spaces" Canad. J. Math. , 3 (1951) pp. 175–186
[a2]	D.K. Burke, "Covering properties" J. Barwise (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. Chapt. 9; pp. 347–422
[a3]	"Generalized metric spaces" J. Barwise (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 423–501
[a4]	P.S. Aleksandrov, V.I. Ponomarev, "Some classes of -dimensional spaces" Sib. Mat. Zh. , 1 (1960) pp. 3–13 (In Russian)
[a5]	J.-I. Nagata, "Modern general topology" , North-Holland (1985)