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In topology, I saw a result about local connectedness that says: If $X$ is a locally connected space and $f:X \longrightarrow Y$ is a continuous open and surjective map then $Y$ is also locally connected. The proof is indeed straightforward, but I found on a pdf of internet that the result is also valid for closed, continuous and surjective maps. I tried to take complements and proceed identically to the argument for the one that works with open maps but I got stucked because $f[A\backslash B] \neq f[A]\backslash f[B]$. So, my question is:

Is the result really valid for closed maps? If so, any hint to prove it is very wellcome.

Thanks!

TransfiniteGuy
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1 Answers1

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An open continuous surjective map and a closed continuous surjective map are both special cases of a quotient map $f: X \to Y$: a (surjective) map such that $$\forall O: f^{-1}[O] \text{ open in } X \iff O \subseteq Y \text{open in } Y$$

And those maps preserve local connectedness, as I showed here.

Henno Brandsma
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