Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
$\begingroup$ $\endgroup$
12 
- 2$\begingroup$ Not an answer. Consider the set $C$ in the plane defined as the union of the vertical segment $S:=\{0\}\times [0,1]$ with the graph of the function $$f:(0,1]\to\mathbb{R},$$ $f(x)=\sin(1/x)$. The set $C$ is closed, connected and I do not believe that it is path connected (Better check my claim). If my claim is correct, then the spaces you are enquiring are pretty weird: no closed subset of such a space can be homeomorphic to a closed cube of dimension $\geq 2$. With the caveat that my claim could be horribly wrong. $\endgroup$Liviu Nicolaescu– Liviu Nicolaescu2020-08-13 18:49:07 +00:00Commented Aug 13, 2020 at 18:49
- 2$\begingroup$ @LiviuNicolaescu Your space is not path connected. $\endgroup$Piotr Hajlasz– Piotr Hajlasz2020-08-13 19:08:10 +00:00Commented Aug 13, 2020 at 19:08
- 3$\begingroup$ @LiviuNicolaescu Unless I'm mistaken, the interval $[0,1]$ has this property. Of course every compact totally disconnected space is also an example (for trivial reasons). $\endgroup$Denis Nardin– Denis Nardin2020-08-13 21:09:56 +00:00Commented Aug 13, 2020 at 21:09
- 2$\begingroup$ Wouldn't 1-dimensional CW complexes work likewise? $\endgroup$Nate Eldredge– Nate Eldredge2020-08-13 21:16:17 +00:00Commented Aug 13, 2020 at 21:16
- 2$\begingroup$ @DenisNardin More generally, every hereditarily locally connected (i.e. every connected subspace is locally connected) Polish space has this property. $\endgroup$D.S. Lipham– D.S. Lipham2020-08-13 21:16:48 +00:00Commented Aug 13, 2020 at 21:16
| Show 7 more comments