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If $X$ is a compact metric space, then $X$ is connected.
I have been attempting to prove the following statement for a while now and I was wondering if anyone would be able to give me a push in the right direction.
If $X$ is a compact metric space, then $X$ is connected.
My attempt: I was wondering if it is sufficient to show that $X$ is not disconnected. To be more explicit I was trying to use the fact that $X$ is connected if and only if each point $x\in X$ has a neighborhood basis for which each of the neighborhoods is connected.
Thus to show that $X$ is not disconnected we seek to show that each point $x\in X$ can be de formed from the rest of $X$. To do this I had thought I would find a neighborhood of $x$ that misses $x$ and then use compactness to find a smaller neighborhood of $x$ that is disjoint from the original neighborhood.
To be more specific to the situation I am working with, I was thinking about using the fact that $X$ is compact. If one picked some point $x\in X$ and a neighborhood of $x$ that missed $x$ the compactness of $X$ would tell us that there is some $N$ such that $X\setminus N \supset U$ for all open sets $U$ in $X$.
We can then attempt to find a $K$ such that $X\setminus K \supset V$ for all open sets $V$ in $X$.
Again, I would like to see a complete answer, but also a push in the right direction.
A:
Suppose there is a partition $A\cup B$ of the space $X$ where $A,B$ are disjoint and
$
A
eq \emptyset,\ \ \ B
eq \emptyset.
$
Consider a point $a\in A$ and a point $b\in B$. Then there are two open 0b46394aab
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