54
$\begingroup$

The Section on Covering Maps in John Lee's book "Introduction to Smooth Manifolds" starts like this:

Suppose $\tilde{X}$ and $X$ are topological spaces. A map $\pi : \tilde{X} \to X$ is called a covering map if $\tilde{X}$ is path-connected and locally path connected, ... (etc).

I hope this question is not too dumb, but how can a space be path connected, but not locally path connected ?

EDIT: I am aware of spaces that are locally path-connected yet not path-connected, but I cannot come up with a space that is path - connected yet not locally path connected.

$\endgroup$
5
  • 22
    $\begingroup$ See here $\endgroup$
    – David Mitra
    Apr 22, 2012 at 19:51
  • $\begingroup$ This is a great comment. $\endgroup$
    – Kerry
    Apr 22, 2012 at 19:53
  • $\begingroup$ @DavidMitra: WOW .. Topology always amazes me, there are so many things that I learn from these counterexamples .. many thanks for pointing me to the link!! $\endgroup$
    – harlekin
    Apr 22, 2012 at 19:54
  • 3
    $\begingroup$ I am to unsure to answer: "because the path witnessing path connectedness might have to pass though a specific point (or be otherwise constrained)". There are other examples. From Steen and Seebach's Counterexamples in Topology: The Alexandroff Square (ex 101), The Extended Topologist's Sine Curve (ex 118), The Closed Infinite Broom (ex. 120), and the Integer Broom (ex 121). $\endgroup$
    – David Mitra
    Apr 22, 2012 at 20:01
  • $\begingroup$ For another example (a subspace of $\mathbb R^2$ which is path-connected and locally connected but not locally path-connected, due to M. Shimrat) see math.stackexchange.com/questions/1492772/… $\endgroup$
    – bof
    yesterday

3 Answers 3

Reset to default
81
$\begingroup$

One counterexample is a variant on the famous topologist's sine curve.

Consider the graph of $y = \sin(\pi/x)$ for $0<x<1$, together with a closed arc from the point $(1,0)$ to $(0,0)$:

enter image description here

This space is obviously path-connected, but it is not locally path-connected (or even locally connected) at the point $(0,0)$.

$\endgroup$
7
  • 1
    $\begingroup$ what is the fundamental group of the picture above? $\endgroup$
    – Ronald
    Jul 31, 2013 at 15:35
  • 2
    $\begingroup$ @Ronald The picture above is simply connected, so its fundamental group is trivial. $\endgroup$
    – Jim Belk
    Nov 14, 2013 at 14:18
  • 1
    $\begingroup$ Question to Jim: why is your space simply connected? it looks like the circle, which is not simply connected... $\endgroup$
    – Hila
    May 27, 2014 at 15:56
  • 26
    $\begingroup$ @Hila It's simply connected because it isn't possible for a path to make it around the "circle". (The sine wave portion is an impassible road block.) $\endgroup$
    – Jim Belk
    May 27, 2014 at 19:10
  • 5
    $\begingroup$ Some people call this space Warsaw circle. Google Images, Google, StackExchange. $\endgroup$
    – Martin Sleziak
    May 8, 2015 at 9:33
8
$\begingroup$

You should consider the opposite question, that how a space could be locally path connected, but not path connected. And this should be simple: consider the union of two open disks.

$\endgroup$
5
  • $\begingroup$ I think harlekin's point, then, is why both hypotheses are being made. Why not just say $\widetilde{X}$ is locally path-connected? $\endgroup$
    – KCd
    Apr 22, 2012 at 19:48
  • $\begingroup$ Thanks for your comment! I have added my post to clarify what confuses me - in my topology course I have seen spaces that are locally path-connected yet not path-connected, but what I have trouble with is coming up with a path-connected space that is not locally path-connected. Yet this is what Lee's opening part of the definition of a covering map suggests exists - I suppose .. $\endgroup$
    – harlekin
    Apr 22, 2012 at 19:49
  • 1
    $\begingroup$ $KCd: I understand. If I am not being mistaken I think Hatcher's book has some discussion relevant to this. $\endgroup$
    – Kerry
    Apr 22, 2012 at 19:51
  • $\begingroup$ Ok I shall have a look at Hatcher's book as well, I am currently reading about the Comb space, as suggested by David, but thanks a lot for your suggestion ! $\endgroup$
    – harlekin
    Apr 22, 2012 at 19:55
  • $\begingroup$ On page 63 he commented that if the space is both path-connected and locally path-connected, then components are the same as path components, which simplifies his discussion on the Galois correspondence on the covering space. $\endgroup$
    – Kerry
    Apr 22, 2012 at 19:59
4
$\begingroup$

$\pi$-Base, an online version of Steen and Seebach's Counterexamples in Topology, lists the following spaces as path-connected but not locally path-connected. You can view the search result for more information about these spaces.

Alexandroff Square

Extended Topologist’s Sine Curve

The Closed Infinite Broom

The Integer Broom

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .