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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?

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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…

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  • $\begingroup$ Oh, I see, its really simple! Thank you! $\endgroup$ – Lucas Corrêa Dec 26 '18 at 19:58

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