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The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501)

Every Polish space (i.e. every separable completely metrizable space) is the image of the Baire space $\mathbb{N}^\mathbb{N}$ by a continuous and closed map.

I was surprised not to find this result (even disguised as an exercise) in Kechris' Classical Descriptive Set Theory, since it feels like a standard result in descriptive set theory. I would like to know if there is a (relatively) recent book or general treatise that contains this result.

Thanks!

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  • $\begingroup$ Have you checked Kechris 7.9 Theorem? $\endgroup$
    – Burak
    Commented Nov 29, 2022 at 16:53
  • $\begingroup$ @Burak, yes. The continuous surjection he builds is bijective, but (therefore) not closed in general. $\endgroup$
    – Lorenzo
    Commented Nov 29, 2022 at 17:03
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    $\begingroup$ @Burak Exercise 7.14 is about every non-empty Polish space being an open image of the Baire space. Kechris mentions the result about closed image - but without a proof: "In R. Engelking [1969] it is shown that X can also be obtained as a continuous and closed image of $\mathcal N$." $\endgroup$
    – Martin Sleziak
    Commented Dec 2, 2022 at 7:06

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