# Path connectedness and locally path connected

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.

• See here Apr 22, 2012 at 19:51
• This is a great comment. Apr 22, 2012 at 19:53
• @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!! Apr 22, 2012 at 19:54
• 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). Apr 22, 2012 at 20:01

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

• what is the fundamental group of the picture above? Jul 31, 2013 at 15:35
• @Ronald The picture above is simply connected, so its fundamental group is trivial. Nov 14, 2013 at 14:18
• Question to Jim: why is your space simply connected? it looks like the circle, which is not simply connected...
– Hila
May 27, 2014 at 15:56
• @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.) May 27, 2014 at 19:10
• Some people call this space Warsaw circle. Google Images, Google, StackExchange. May 8, 2015 at 9:33

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.

• I think harlekin's point, then, is why both hypotheses are being made. Why not just say $\widetilde{X}$ is locally path-connected?
– KCd
Apr 22, 2012 at 19:48
• 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 .. Apr 22, 2012 at 19:49
• $KCd: I understand. If I am not being mistaken I think Hatcher's book has some discussion relevant to this. Apr 22, 2012 at 19:51 • Ok I shall have a look at Hatcher's book as well, I am currently reading about the Comb space, as suggested by David, but thanks a lot for your suggestion ! Apr 22, 2012 at 19:55 • On page 63 he commented that if the space is both path-connected and locally path-connected, then components are the same as path components, which simplifies his discussion on the Galois correspondence on the covering space. Apr 22, 2012 at 19:59$\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