Metrizable space

In the sub-area of topology of mathematics , a metrizable space is a topological space with additional special properties.

Since the metric spaces are special cases of the topological spaces, it makes sense to ask when a topological space can be metrised , that is, which additional requirements a topological space has to meet in order for there to be a metric that induces the topology. This article gives an overview of necessary and sufficient conditions for metrizability , which are explained in more detail in the articles to which reference is made from here. Sentences that weak sufficient conditions or equivalent conditions to formulate Metrisierbarkeit be in the literature as Metrisationssätze referred.

Necessary conditions

Every topological property that metric spaces always fulfill naturally represents a necessary condition for the metrizability of any topological space. Of particular interest, however, are properties that “bring the space close to metrizability”.

Equivalent condition

Nagata-Smirnow's metrization theorem: A topological space is metrizable if and only if it is a regular Hausdorff space and has a σ-locally finite basis . (See Bing-Nagata-Smirnov theorem for a more detailed discussion .)

Examples, construction of a metric

The easiest way to construct the metric is when the topological space is a finite product of metric spaces . You can then simply add the metrics, for example: ${\ displaystyle X}$${\ displaystyle (M_ {i}, d_ {i}); \; 1 \ leq i \ leq n}$

${\ displaystyle d ((x_ {1}, x_ {2}, \ dotsc, x_ {n}), (y_ {1}, y_ {2}, \ dotsc, y_ {n})) = d_ {1} (x_ {1}, y_ {1}) + d_ {2} (x_ {2}, y_ {2}) + \ dotsb + d_ {n} (x_ {n}, y_ {n})}$

One can proceed in a similar way if the topological space is a countable product of metric spaces . Then one must force the convergence of the “infinite sum” through a positive sequence and, if necessary , replace the metrics d i with topologically equivalent metrics bounded by a common limit. The definition does both: ${\ displaystyle X}$${\ displaystyle (M_ {i}, d_ {i}); \; i \ in \ mathbb {N}}$

${\ displaystyle d ((x_ {i}), (y_ {i})) = \ sum \ limits _ {i = 0} ^ {\ infty} 2 ^ {- i} {\ frac {d_ {i} ( x_ {i}, y_ {i})} {1 + d_ {i} (x_ {i}, y_ {i})}}}$

Counterexamples

• The product topology of at least two-point metric spaces cannot be metrized if the index set can be overcounted.${\ displaystyle \ textstyle X = \ prod _ {i \ in I} M_ {i}}$${\ displaystyle I}$
• The first non-countable ordinal number , provided with its order topology, is not paracompact and therefore cannot be metrised.${\ displaystyle \ Omega _ {0}}$

Individual evidence

1. ^ Schubert: Topology. 1975, p. 97.
2. Eduard Čech : On Bicompact Spaces . In: Annals of Mathematics . Vol. 38, No. 4, 1937, pp. 823-844. doi : 10.2307 / 1968839 .
3. Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. 1970, p. 180 .