spaces homeomorphic to Baire space
Baire space, 𝒩≡ℕℕ, is the set of all functions x:ℕ→ℕ together with the product topology. This is homeomorphic to the set of irrational numbers in the unit interval, with the homeomorphism f:𝒩→(0,1)∖ℚ given by continued fraction expansion
f(x)=1x(1)+1x(2)+1⋱. |
Theorem 1.
Let I be an open interval of the real numbers and S be a countable dense subset of I. Then, I∖S is homeomorphic to Baire space.
More generally, Baire space is uniquely characterized up to homeomorphism by the following properties.
Theorem 2.
A topological space X is homeomorphic to Baire space if and only if
-
1.
It is a nonempty Polish space.
-
2.
It is zero dimensional (http://planetmath.org/ZeroDimensional).
-
3.
No nonempty open subsets are compact.
In particular, for an open interval I of the real numbers and countable dense subset S⊆I, then I∖S is easily seen to satisfy these properties and Theorem 1 follows.
Title | spaces homeomorphic to Baire space |
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Canonical name | SpacesHomeomorphicToBaireSpace |
Date of creation | 2013-03-22 18:46:48 |
Last modified on | 2013-03-22 18:46:48 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 7 |
Author | gel (22282) |
Entry type | Theorem |
Classification | msc 54E50 |
Related topic | PolishSpace |
Related topic | InjectiveImagesOfBaireSpace |