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Is a polish space (complet metric separable topological space) sigma compact ?

Thanks and regards.

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Not in general - indeed baire space $\omega^\omega$ is not $\sigma$-compact. cf. this answer.

I hope this helps ^_^

HallaSurvivor
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  • the irrationals as a subspace of the real line are not $F_\sigma$ (the rationals are not $G_\delta$), so the irrationals are not $\sigma$-compact. – Mirko Nov 01 '19 at 00:39
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Another example, Hilbert space $l^2$. Complete separable metric, but not sigma compact.

Steven Clontz
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GEdgar
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  • I thought so too at first, but it actually is $\sigma$-compact. Just start with the 1-ball in one dimension and then - step by step - increase dimension and radius. – amsmath Oct 31 '19 at 15:30
  • Wrong. Think about it. The vector $(1,1/2,1/3,1/4,\cdots)$ is not in your union. Hilbert space is not equal to $$\bigcup_{n=1}^\infty \mathbb R^n$$ – GEdgar Oct 31 '19 at 19:40
  • You're right. I was wrong. But good that we talked about it. ;-) – amsmath Oct 31 '19 at 19:44
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    @amsmath in an infinite dimensional TVS a compact subset has empty interior, and a $\sigma$-compact TVS is thus meagre (e.g. all eventually $0$ sequences in the sup norm is an example of it) so cannot be Polish. – Henno Brandsma Oct 31 '19 at 21:40
  • Thank you, why $\mathcal{l}^{2}$ is not sigma compact ? –  Nov 01 '19 at 08:24
  • @HennoBrandsma Is there an easy proof for normed spaces? Is it possibly true that a compact set is always contained in a finite-dimensional subspace? – amsmath Nov 01 '19 at 13:02
  • @amsmath ... Is it possibly true that a compact set is always contained in a finite-dimensional subspace? no. For example, a convergent sequence, together with its limit, is a compact set. Perhaps you can find an example like that in Hilbert space not contained in any finite-dimensional subspace. WARNING ... new questions should not be posted as comments. – GEdgar Nov 01 '19 at 14:48
  • @CechMS because it’s Polish it cannot be $\sigma$-compact. – Henno Brandsma Nov 01 '19 at 20:59
  • why ? What is the proof ? –  Nov 02 '19 at 09:10