TITLE: Path-connected and locally connected space that is not locally path-connected
QUESTION [61 upvotes]: I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies path-connectedness.), we can draw this diagram:
+--------------------------+
|Connected                 |
|             1      +-----+----------------------------+
|                    |  3  |           Locally connected|
|   +----------------+-----+     6                      |
|   |Path-connected  |  4  |                            |
|   |                +-----+------------------------+   |
|   |    2           |  5  |  Locally path-connected|   |
+---+----------------+-----+                        |   |
                  8  |           7                  |   |
                     +------------------------------+---|

So, I want to find examples of all these 8 categories, but I can't find an example for 4.

The topologist's sine curve
The comb space
The ordered square
See below
The real line
The disjoint union of two spaces of the 3rd type
$[0,1] \cup [2,3]$
The rationals

Actually there is an answer that gives an example of type 4, but there isn't any explanation. Can anyone please explain it (why it is not locally path-connected, to be specific) or give another example?

REPLY [15 votes]: A very nice example for (4) is a "long circle".  That is, let $L$ be the closed long line, with endpoints $0$ and $\omega_1$, and adjoin to $L$ a path from $0$ to $\omega_1$ to get a space $S$ (or equivalently, you could just identify $0$ and $\omega_1$ together).  Then $S$ is path-connected and locally connected (and also compact and Hausdorff), but it is not locally path-connected at $\omega_1$.