Space Polish

A space métrisable at countable base (or separable, that returns to same for a space métrisable) is a Polish space if its topology can be defined by a distance which in fact a complete Espace. All compact Space métrisable, any closed or opened subspace of a Polish space, very produced countable of Polish spaces, all separable Espace of Banach is a Polish space.

Examples

Thus all usual spaces of the analysis or the functional analysis are Polish.
Any space locally compact at countable base is Polish (it is open in its Compactifié d' Alexandrov).
But there exist many interesting Polish spaces in which very compact is of empty interior, for example separable spaces of Banach of infinite size (because of the theorem of Riesz)
or fundamental Polish space N^N (often called the space of Baire , from where ambiguity with Espace of Baire) which, except for a homeomorphism, is the only completely discontinuous Polish space in which very compact is of empty interior.
To note that] 0,1 is a Polish space, moreover homeomorphic with ''' R ''', but is not complete for the usual distance.

Properties

In a general way, a subspace of a Polish space is itself Polish if it is intersection of a succession of open of space (this type of part has a small name: it is said that it is a G_δ ; to see Class of Baire). Thus Polish spaces are identifiable with the G_δ Cube of Hilbert.
Polish spaces are spaces of Baire. In other words, in a Polish space the complementary one to a thin part contains a dense G_δ .
In particular, if one removes of a Polish space P without isolated points a countable unit D , it remains a dense subspace Polish J of which very compact is of empty interior; if P is an interval of R and D the whole of the rational contents in P , which remainder is moreover more completely discontinuous and thus homeomorphic with N^N (called blow often space of irrational the ; for P =] 0,1 decomposition in [[algorithm of Euclide|fraction continuous (E) E] gives a homeomorphism between J and N^N ).
All measurement μ (limited or σ-finished) on the Tribu borélienne of a Polish space is Intérieurement regular relative with the compact ones. In other words, for all borélien B one has $\ driven (driven B)= \ sup \ {\ (K), \ K \ \ mbox {compact included in} \ B \}$ . Consequently, all borélien is the meeting of a countable meeting the compact ones (small name: K_σ ) and of a μ-negligible unit.
By what precedes one deduces that is the meeting of a thin K_σ and of a G_δ Négligeable for the measurement of Lebesgue. It is a species of paradox, the thin whole in a space of Baire being “negligible” from the topological point of view.

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