Example of a surjective local homeomorphism that is not a covering?

Question

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering?

I don't think this is the same question as here? There's different conditions?

It's not quite the same question, but it has the same answer :) — Ben, Jul 15 '15 at 03:27

score 11 · Accepted Answer · edited Jul 14 '15 at 16:23

11

Restrict the usual covering projection from the real line to the circle(that is $x$ goes to $e^{2\pi ix}$) on the positive side of the line.

$p:R_+\rightarrow S^1$ defined as $p(x)=(\cos2\pi x,\sin2\pi x)$

This is a local homeomorphism but not covering projection. As the point $(1,0)$ on the circle is not evenly covered by the projection map.

edited Jul 14 '15 at 16:23

Stefan Hamcke

28,621

answered Jul 14 '15 at 15:49

Ripan Saha

1,901

What makes you say $(0,1)$ is not evenly covered? – Clive Newstead Jul 14 '15 at 15:59
sorry it would be (1,0).It is a typo. – Ripan Saha Jul 14 '15 at 16:00
Could you provide a bit more detail? – user166854 Jul 14 '15 at 16:03
Why got this answer downvoted? – Stefan Hamcke Jul 14 '15 at 16:11
2

@user141494 inverse of any open set U containing the point (1,0) will look line open intervals $V_a$containing the integer points on $R^+$ ,as all the integers on $R^+$ maps to (1,0) under the projection map. By the definition of covering maps these $V_a$ should be homeomorphic to U when you restricts to $V_a$'s.But in this case open interval say $V_0$ containing 0 is not homeomorphic to U, as restriction map from $V_0$ to U is not surjective. So the above mentioned map is not a covering map. – Ripan Saha Jul 14 '15 at 16:50
2

I think you need to take $R_+$ without $0$, otherwise $p$ is not an open map. – Stefan Hamcke Jul 14 '15 at 18:52
@StefanHamcke yes, you are right....Thanks – Ripan Saha Jul 14 '15 at 19:01
@RipanSaha I don't understand why (1,0) is not evenly covered by the exponential. Can you help me? – Ava3141 Mar 17 '21 at 17:16

Example of a surjective local homeomorphism that is not a covering?

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