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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.

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    See [here](http://en.wikipedia.org/wiki/Comb_space)2012-04-22
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    This is a great comment.2012-04-22
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    @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!!2012-04-22
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    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).2012-04-22

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

Consider the graph of $y = \sin(\pi/x)$ for $0, 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 $(0,0)$.

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    what is the fundamental group of the picture above?2013-07-31
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    @Ronald The picture above is simply connected, so its fundamental group is trivial.2013-11-14
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    Question to Jim: why is your space simply connected? it looks like the circle, which is not simply connected...2014-05-27
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    @Hila Jim never claimed it to be simply connected. You are the first to bring up the term "simply connected" here.2014-05-27
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    @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.)2014-05-27
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    Some people call this space *Warsaw circle*. [Google Images](http://images.google.com/images?q=warsaw+circle), [Google](http://google.com/search?q="warsaw+circle"), [StackExchange](http://google.com/search?q="warsaw+circle"+site:stackexchange.com+OR+site:mathoverflow.net).2015-05-08
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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.

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    I think harlekin's point, then, is why both hypotheses are being made. Why not just say $\widetilde{X}$ is locally path-connected?2012-04-22
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    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 ..2012-04-22
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    $KCd: I understand. If I am not being mistaken I think Hatcher's book has some discussion relevant to this.2012-04-22
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    Ok I shall have a look at Hatcher's book as well, I am currently reading about the Comb space, as suggested by David, but thanks a lot for your suggestion !2012-04-22
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    On page 63 he commented that if the space is both path-connected and locally path-connected, then components are the same as path components, which simplifies his discussion on the Galois correspondence on the covering space.2012-04-22
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$\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