spaces homeomorphic to Baire space
Baire space, , is the set of all functions
together with the product topology. This is homeomorphic to the set of irrational numbers in the unit interval, with the homeomorphism given by continued fraction
expansion
Theorem 1.
Let be an open interval of the real numbers and be a countable
dense subset of . Then, is homeomorphic to Baire space.
More generally, Baire space is uniquely characterized up to homeomorphism by the following properties.
Theorem 2.
A topological space 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 of the real numbers and countable dense subset , then is easily seen to satisfy these properties and Theorem 1 follows.