Baire room (special)

This article deals with the Polish area . For spaces to which Baire's theorem applies, see Baire space (general) . ${\displaystyle {\mathcal {N}}:=\mathbb {N} ^{\mathbb {N} }}$

The Baire space (after the French mathematician René Louis Baire ) is a topological space with special significance for descriptive set theory , since many theorems that can be proven for the Baire space refer to general Polish spaces , for example on or the Hilbert cube transferred immediately. In proofs, the Baire space can be easier to handle than, say, the space of real numbers . The basic set of Baire space is the set of all sequences of natural numbers ( ), and the topology of Baire space (the set of " open " sets of Baire space) consists of those sets of such sequences that consist of the union of sets Sequences with a common prefix are. ${\displaystyle {\mathcal {N}}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle [0,1]^{\mathbb {N} }}$ ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$

The topology of the Baire space

The topology of the Baire-space can be used as the countable product topology of the discrete space over defining what is equivalent to the above intuitive definition of open sets, the sets of sequences with a common prefix thereby form a base of the topology. What has to be shown is that the Baire room is actually a Polish room. Since the discrete topology can be metrised , the countable product can also be metrised; a simple metric can also be specified in concrete terms: ${\displaystyle \mathbb {N} }$

${\displaystyle d\colon {\mathcal {N}}\times {\mathcal {N}}\to \mathbb {R} ,(x,y)\mapsto {\frac {1}{2^{n(x,y)}}}}$,

where the first position denotes where the sequences and differ. It's even an ultrametric . The Baire space is separable , since the set of all sequences that take on the value zero from a certain position onwards can be counted and is close to . The completeness can be shown analogously to the real numbers, since the Baire space allows an interval nesting in a very natural way. ${\displaystyle n(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle {\mathcal {N}}}$

Significance for descriptive set theory

The universal usability of the Baire space is that every Polish space is the image of a continuous image from the Baire space, i.e. This means that there is a continuous surjection of in every Polish area. ${\displaystyle {\mathcal {N}}}$

Properties of projective sets are particularly easy to show in Baire space, since every countable product of Baire space is homeomorphic to Baire space itself; H. is homeomorphic to for , and . This makes the definition of projective sets particularly easy. In addition, every analytic set in Baire space is a projection of a closed set, while projections of sets must be considered for real numbers or Cantor space . ${\displaystyle {\mathcal {N}}}$ ${\displaystyle {\mathcal {N}}^{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle {\mathcal {N}}^{\mathbb {N} }}$ ${\displaystyle \mathbb {N} \times {\mathcal {N}}}$ ${\displaystyle \Pi _{2}^{0}}$

Properties of the topology

Every point in Baire space has the set of all sets as a countable environment basis , whereby here denotes the set of all points in Baire space so that the first digits coincide with those of . This allows a natural characterization of the continuity of a function at one point : is continuous at the point when each one exists, so that the first points of the first points of determine. ${\displaystyle U_{n}(x)}$ ${\displaystyle U_{n}(x)}$ ${\displaystyle n}$ ${\displaystyle x}$ ${\displaystyle f\colon {\mathcal {N}}\to {\mathcal {N}}}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle m}$ ${\displaystyle x}$ ${\displaystyle n}$ ${\displaystyle f(x)}$

Similarly, the convergence can be characterized a series of points of Baire-space: A sequence converges if and only if each one does so from th follower the first places always the same. This distinguishes the Baire space from the real numbers, in which edge cases occur in which this property is violated (the sequence 0.9, 0.99, 0.999, ... converges to one). ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle m}$ ${\displaystyle n}$

The Baire space is - like every ultrametrizable space - totally incoherent , every subspace with at least two elements can be divided into two disjoint open sets. This is very easy to see for open sets: If you represent an open set as a set of prefixes, which is always possible because the sets of sequences with a certain prefix form a basis, the open set can be added by adding additional characters split a prefix into disjoint open subsets. An example: The set of all sequences starting with zero in Baire space is open; it can be broken down into the open set of sequences starting with zero followed by an even number and the open set of sequences beginning with zero followed by an odd number split up. The property is carried over to the subspace topology for non-open sets.

There are no isolated points in baire space ; H. it is a perfect Polish room .

The Baire space can also be embedded in the real numbers : Using the continued fraction expansion it can be shown that it is homeomorphic to the subspace of the irrational numbers . The function

${\displaystyle \iota \colon {\mathcal {N}}\to \mathbb {R} ,s\mapsto 1+s(0)+{\cfrac {1}{1+s(1)+{\cfrac {1}{1+s(2)+{\cfrac {1}{1+s(3)+{\cfrac {1}{\;\,\ddots }}}}}}}}}$(where the zero contains) ${\displaystyle \mathbb {N} }$

is a homeomorphism on your image that is just greater than the set of irrational numbers . This in turn is homeomorphic to the irrational numbers in the interval by means of homeomorphism . Thus, the irrational numbers are countable topological sums of sets homeomorphic to Baire space and thus also homeomorphic to Baire space. It should be noted that this subspace of the real numbers is not closed and therefore the usual metric inherited from the real numbers is not complete on it. ${\displaystyle 1}$ ${\displaystyle x\mapsto \textstyle {\frac {1}{x}}}$ ${\displaystyle (0,1)}$

To universality

As noted, a fundamental property of the Baire area is that every Polish area is a continuous image of the Baire area. So be a Polish room. Now open subsets can be constructed for all finite words with the following properties: ${\displaystyle X}$ ${\displaystyle w\in \mathbb {N} ^{*}}$ ${\displaystyle U_{w}}$

• ${\displaystyle U_{\varepsilon }=X}$
• ${\displaystyle {\overline {U_{wi}}}\subset U_{w}}$
• ${\displaystyle U_{w}=\bigcup _{i\in \mathbb {N} }U_{wi}}$
• ${\displaystyle \operatorname {diam} (U_{w})\to 0}$for . ${\displaystyle |w|\to \infty }$

For this one needs the separability: for each there is a countable dense subset, sufficiently small open spheres around these then fulfill the desired properties. Now, define with , wherein the prefix of the length of call. Due to the completeness of the space, the interval nesting principle is clearly defined. is continuous because it receives convergence: Points of the Baire space with a common prefix are mapped to points in , so that the image sequence of a convergent sequence converges to the image of the limit value. There is even a continuous bijection from a closed subspace to (note that this does not have to be a homeomorphism, but can also coarsen the topology - the category of Polish spaces is not balanced - in particular, such a subspace is totally disconnected, which is why it is not can be homeomorphic to the real numbers). Instead of open sets , choose -Sets , which, in addition to the above conditions, are also pairwise disjoint at each level, i.e. for . You get such a thing through . If one now restricts to the points for which is not empty and is therefore just the same , one obtains the desired continuous bijection. ${\displaystyle U_{w}}$ ${\displaystyle f\colon {\mathcal {N}}\to X}$ ${\displaystyle f(s)\in \textstyle \bigcap _{n\in \mathbb {N} }U_{s|n}}$ ${\displaystyle s|n}$ ${\displaystyle s}$ ${\displaystyle n}$ ${\displaystyle f(s)}$ ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle U_{p}}$ ${\displaystyle X}$ ${\displaystyle U_{w}}$ ${\displaystyle \Delta _{2}^{0}}$ ${\displaystyle W_{w}}$ ${\displaystyle W_{wi}\cap W_{wj}=\emptyset }$ ${\displaystyle i\neq j}$ ${\displaystyle W_{wi}=W_{w}\cap U_{wi}\setminus \textstyle \bigcup _{j<i}U_{wj}}$ ${\displaystyle f}$ ${\displaystyle s}$ ${\displaystyle \textstyle \bigcap _{n\in \mathbb {N} }W_{s|n}}$ ${\displaystyle f(s)}$

Retracts

In the Baire room, every closed sub-room (these are especially Polish) is also a retract . To prove this, one covers the subspace as before with open sets with the additional condition that for all . The mapping constructed as above is then again continuous and each element of is a fixed point. Thus it is a retraction and the subspace is a retract. The restriction provides that every closed subspace of a closed subspace is also a retract of the latter. Conversely, as in every Hausdorff area, each retract must be closed, i.e. H. the retracts are thus fully characterized. ${\displaystyle A\subset {\mathcal {N}}}$ ${\displaystyle U_{w}}$ ${\displaystyle a\in A}$ ${\displaystyle a\in U_{a|n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a}$

See also

• Cantor room

Individual evidence

1. ^ Donald A. Martin : Classical Descriptive Set Theory. In: Jon Barwise (Ed.): Handbook of Mathematical Logic (= Studies in Logic and the Foundations of Mathematics. Vol. 90). North-Holland, Amsterdam et al. 1977, ISBN 0-7204-2285-X , pp. 783-818, here p. 785: "We do not use the real line, because that space is slightly awkward to use."
2. ↑ ^{a b c d} David Marker: Descriptive Set Theory . 2002, (Lecture notes; PDF; 643 kB).
3. ^ Donald A. Martin: Classical Descriptive Set Theory. In: Jon Barwise (Ed.): Handbook of Mathematical Logic (= Studies in Logic and the Foundations of Mathematics. Vol. 90). North-Holland, Amsterdam et al. 1977, ISBN 0-7204-2285-X , pp. 783-818, here p. 790.
4. ↑ Oliver Deiser: Real Numbers. The classical continuum and the natural consequences. Springer, Berlin et al. 2007, ISBN 978-3-540-45387-1 , 2nd section 1: Introduction to the bar room.
5. ↑ Oliver Deiser: Real Numbers. The classical continuum and the natural consequences. Springer, Berlin et al. 2007, ISBN 978-3-540-45387-1 , pp. 303, 305 ff.