Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

Asked Jun 19 '15 at 14:35

Active Jun 19 '15 at 15:35

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Result: If $X$ is a complete separable metric space then there is a $E \subset [0,1]^{\mathbb{N}}$ such that $X$ is homeomorphic to $E$ ($E$ is a $G_\delta$ set - is the intersection of denumerable open sets)

This result is used to prove theorem 2.3 of [PARTHASARATHY, 1967, Probability measures on metric spaces, pg 18]. and it is quoted as a well know result.

1) How to prove this result?

This result is to be found (according the author) in Kuratowski's 1933 book "Topologie I" pages 215-216. But I couldn't find it.

2) Could you indicate a good book where this result is to be found?

edited Jun 19 '15 at 15:35

Martin Sleziak

53,687

asked Jun 19 '15 at 14:35

Conrado Costa

6,071

1

You can find some questions on this site which are closely related. For example: Polish Spaces and the Hilbert Cube, Urysohn's metrization theorem and Borel image. – Martin Sleziak Jun 19 '15 at 15:29
If you simple google for some reasonable phrases (like "hilbert cube" separable metric or separable metric embedded) you might be able to find a few references by yourself. – Martin Sleziak Jun 19 '15 at 15:31

Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

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