Polish area

In the sub-area of topology of mathematics , a Polish space is a separable and completely metrizable topological space .

Here, fully metrizable means that there is a metric on which induces the topology and is at the same time complete , that is, that every Cauchy sequence converges with respect to . (A metric induced topology on if we are open quantities of by open balls with respect to explain.) Note that the completeness depends on the metric: Is the space with respect to a metric complete, there may be other metrics that same Generate topology and are not complete. It is required here that there is at least one complete metric that generates the topology. ${\ displaystyle d}$${\ displaystyle X}$${\ displaystyle d}$${\ displaystyle d}$${\ displaystyle X}$${\ displaystyle X}$${\ displaystyle d}$

A topological space is called separable if there is a countable and dense subset , that is, it is equal to the set of natural numbers and it applies . This property restricts the size of Polish spaces, so they are also accessible to methods of mass theory . ${\ displaystyle X}$ ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle {\ overline {A}} = X}$

Polish spaces are equally characterized by the fact that they are completely metrizable and their topology has a countable basis .

Separable and completely metrizable topological spaces are in honor of the Polish mathematicians who dealt first with them ( Sierpiński , Kuratowski , Tarski ) polish called. The terminology goes back to Nicolas Bourbaki . Polish spaces are the central subject of investigation in descriptive set theory and play an important role in measurement theory , for example in connection with radon measures .

Effective Polish rooms

An effective Polish space is a Polish space that has a predictable representation. Such spaces are the subject of effective descriptive set theory and constructive analysis .

Formally, an effective Polish space is a Polish space with a metric such that there is a countable dense set that makes the following two relations computable: ${\ displaystyle d}$${\ displaystyle C = (c_ {0}, c_ {1}, \ dotsc)}$${\ displaystyle \ mathbb {N} ^ {4}}$

${\ displaystyle P (i, j, k, m) \ equiv d (c_ {i}, c_ {j}) \ leq {\ frac {m} {k + 1}}}$
${\ displaystyle Q (i, j, k, m) \ equiv d (c_ {i}, c_ {j}) <{\ frac {m} {k + 1}}}$

Examples

1. Any finite or countably infinite discrete space is a Polish space.
2. For each , with its natural topology, there is a Polish space.${\ displaystyle n}$${\ displaystyle \ mathbb {R} ^ {n}}$
3. In general, every separable Banach space is provided with the topology induced by its norm in Polish, for example many function spaces such as the -spaces , the Sobolev spaces or the sequence spaces for finite or common metric spaces of continuous functions.${\ displaystyle L ^ {p}}$ ${\ displaystyle W ^ {k, p}}$ ${\ displaystyle \ ell ^ {p}}$${\ displaystyle p}$
4. Every compact metrizable room is Polish.
5. In general, every locally compact , metrizable space , which can be counted in infinity, is a Polish space.
6. The product of Polish spaces (equipped with the product topology) forms a Polish space if the index set I is finite or countable.${\ displaystyle \ prod _ {i \ in I} X_ {i}}$${\ displaystyle X_ {i}}$
7. The Cantor discontinuum is a Polish space.
8. The set of irrational numbers forms a Polish area. In the usual ("Euclidean") metric (which is defined by ) the irrational numbers are not complete; a sequence of irrational numbers that converges to a rational number is indeed a Cauchy sequence, but has no limit in the space of irrational numbers. However, the irrational numbers are homeomorphic to Baire space , the product of countably many copies of the natural numbers. A complete metric on the irrational numbers can be specified explicitly as follows: if the first terms of the continued fraction expansion of and match, but the -th term does not .${\ displaystyle d (x, y) = | xy |}$${\ displaystyle \ mathbb {N} ^ {\ mathbb {N}}}$${\ displaystyle d (x, y) = {\ tfrac {1} {n + 1}}}$${\ displaystyle n}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle n + 1}$
9. Every closed sub-room of a Polish room is in turn a Polish room.
10. A subspace of a Polish space is itself a Polish space if and only if it is a G δ -set, i.e. the intersection of countably many open subsets in the given topology ( Mazurkiewicz's theorem ).
11. Except for homeomorphism , the Polish spaces are exactly the G δ subsets of the Hilbert cube .${\ displaystyle [0,1] ^ {\ mathbb {N}}}$
12. Every Polish room is an image of a steady surjection from the Baire area. The Baire room is effective just like the Cantor room .

Individual evidence

1. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , chapter 8.1.
2. a b Bauer: Measure and Integration Theory. 1992, p. 178.
3. Querenburg: Set theoretical topology. 1973, p. 148.
4. ^ Bauer: Measure and integration theory. 1992, pp. 178-190.
5. ^ Yiannis N. Moschovakis : Descriptive Set Theory (= Mathematical Surveys and Monographs. Vol. 155). 2nd edition. American Mathematical Society, Providence RI 2009, ISBN 978-0-8218-4813-5 .
6. Querenburg: Set theoretical topology. 1973, p. 149.
7. a b Querenburg: Set theoretical topology. 1973, p. 150.
8. Oliver Deiser: Real Numbers. The classical continuum and the natural consequences. 2nd, corrected and enlarged edition. Springer, Berlin et al. 2008, ISBN 978-3-540-79375-5 , corollary on page 335.