Common
Polish space
Concept in topology
Polish space
In mathematics, a Polish space is a separable, completely metrizable topological space; i.e., a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians, such as Sierpiński, Kuratowski, Tarski and others. Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory. Common examples of Polish spaces are the real line, the Baire space, Cantor spaces and separable Banach spaces. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish: for example, open intervals are Polish. Any two uncountable Polish spaces are Borel isomorphic. In particular, every uncountable Polish space has the cardinality of the continuum. Lusin...
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