Polish space
In subdivision topology mathematics, a Polish space is a separable completely metrizable topological space and.
Thereby completely metrisable means that there is a metric, which induces at the same time and the topology is completed, that is, that each fundamental sequence converges respect. ( A metric induces the topology if we can upon to explain the open sets of by open balls. ) Note that the completeness depends on the metric: Is the space with respect to a metric completely, so there may be other metrics that have the same generating topology, and are not exhaustive. Here, it is required that there is at least a complete metric generating the topology.
A topological space is called separable if there is a countable and dense subset, ie is equal to the powerful set of natural numbers, and it is. This property Polish spaces are limited in their size, they are therefore accessible measure theoretic methods.
Polish spaces are equivalent characterized in that they are completely metrizable and its topology has a countable basis.
Separable and completely metrizable topological spaces are named in honor of the Polish mathematician who first dealt with them ( Sierpiński, Kuratowski, Tarski ), called in Polish. The terminology goes back to Nicolas Bourbaki. Polish spaces are a central issue to the descriptive set theory and play an important role in measure theory, such as those associated with radon measurements.
Effective Polish spaces
An effective Polish space is a Polish space, which has a computable representation. Such premises are the subject of effective descriptive set theory and constructive analysis.
Formally, an effective Polish space is a Polish space with a metric, so there is a countable dense set, which makes the following two relations on computable: