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?