Definition 111.6.26. A topological space $X$ is called connected if it is nonempty and not the union of two nonempty disjoint open subsets. A connected component of $X$ is a maximal connected subset. Any point of $X$ is contained in a connected component of $X$ and any connected component of $X$ is closed in $X$. (But in general a connected component need not be open in $X$.)
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