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Given X, define an equivalence relation on X by setting x~y if there is a connected subspace of X containing both x and y. The equivalence classes are called yhe components of X.

This is what I learned as definition of components. But I think reflexivity of the equivalence relation is unsure. For every x in X, to satisfy the condition x~x, we should show that there is a connected subspace of X containing x. How can I show this?

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The set $\{x\}$ is a connected subspace of $X$ to which $x$ belongs.

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$\{x\}$ is always connected. Just apply the definition of connectedness to see this.

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