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.
$\endgroup$ 43 Answers
$\begingroup$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 $(0,0)$:
This space is obviously path-connected, but it is not locally path-connected (or even locally connected) at the point $(0,0)$.
$\endgroup$ 7 $\begingroup$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$$\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
$\endgroup$
