If a metric space $M$ is the union of path-connected sets $S_{\alpha}$, all of which have the nonempty path-connected set $K$ in common, is $M$ path-connected?
I didnt have a good idea. I know that a union of connected sets sharing a point in common is connected so, I tried to generalize this for path-connected, but I couldnt.
My idea to prove this was take $p \in M$ and define the set
$$X = \{x \in M \mid x\text{ can be connected to }p\text{ by a path countained in }M\}$$
and show that $X = M$. Can someone help me?
Take $p,q\in M$. Choose a point $k\in K$. Then there is a path in $M$ joining $p$ to $k$ and there is a path in $M$ joining $k$ to $q$. So…