Discrete fibers vs local homeomorphism vs covering map

Asked Jul 10 '23 at 22:40

Active Jul 10 '23 at 22:40

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I was wondering about the difference between the following possible conditions on a given continuous map $f : X \rightarrow Y$ between topological spaces:

$f$ has discrete fibers
$f$ is a local homeomorphism
$f$ is a covering map.

As far as I know, 3 implies 1 and 2; however, I am wondering if there are any further implications between these three properties, or if there are counter-examples between them.

Would anyone have any suggestions as to where I can find proofs or counter-examples of these?

asked Jul 10 '23 at 22:40

I.A.S. Tambe

2,903

3

3 implies 2 and 2 implies 1. Neither implication is reversible. – Zhen Lin Jul 10 '23 at 22:53
2

To see that 1 does not implies 2, consider the map $f\colon \mathbb{C} \to \mathbb{C}$ given by $f(z) = z^2$. – Henrique Augusto Souza Jul 10 '23 at 22:55
2

here you get examples of 2 not implying 3: https://math.stackexchange.com/questions/447168/local-homeomorphisms-which-are-not-covering-map – Henrique Augusto Souza Jul 10 '23 at 22:57
Thanks very much for your help! – I.A.S. Tambe Jul 10 '23 at 22:57

Discrete fibers vs local homeomorphism vs covering map

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