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?
