Is a polish space sigma compact?

Question

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

Thanks and regards.

Answer 1

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

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I hope this helps ^_^

Answer 2

Another example, Hilbert space $l^2$. Complete separable metric, but not sigma compact.