# An open connected subset of compact metric connected and locally connected space, is path connected

I need to prove:
If $X$ is compact, metric, connected and locally connected space, and $U$ is open connected subset of $X$, then $U$ is path connected.

Using the following:

a) If $X$ connected, and $F$ is open cover of $X$, then for every $a,b∈X$ exists an $n\in\mathbb{N}$ and open sets $U_1 ,...,U_n∈F$ such that $a∈U_1$ and $b∈U_n$ and $U_i \cap U_j \neq \emptyset \leftrightarrow |j-i|=1$.

b) Every compact metric connected space with more than one point, has at least two non-cut points.

c) If $X$ is a compact metric connected space with exactly two non-cut points, then $X$ is homeomorphic to $[0,1]$.

d) If $X$ is a compact metric connected and locally connected space and $a,b∈X$, then there is a closed connected subset of $X$, where its points, apart from $a$ and $b$, are cut points.

I proved a,b,c, but I have trouble to build the subset in d.
and I can't figure how the theorem follows from a,b,c,d.

can anyone help me?

• Could you spell out what you mean by d)? As it stands, you could just take the empty set as all elements of the empty set are cut-points. – fuglede Oct 17 '14 at 12:01
• @fuglede But the empty set is not connected I think. – Marik S. Oct 17 '14 at 12:11
• @MarikS.: That's a matter of convention. Also, if $X$ itself has no cut-points, the claim is problematic. The importance of $a$ and $b$ is unclear to me as well. – fuglede Oct 17 '14 at 12:14
• a,b,c,d should help me prove the theorem. therefore I think the empty set (if it is connected or not) won't help. The points a,b may be non-cut points (if we want to use c to prove the theorem). – alex kur Oct 17 '14 at 12:21
• What is this question? You proved the Cut Point Non-Existence Theorem as part of a question? And as the next part, the Moore classification of the arc? Yeah right. If you have the fact that a Peano continuum is path-connected then (d) is totally trivial, and if you don't have that fact then you need to prove the Hahn-Mazurkiewicz Theorem among many other facts, which takes pages and pages to do. This is like some sort of 'fake news' of MSE posts haha – John Samples Oct 8 '17 at 3:13