Definition 5.7.1. Let $X$ be a topological space.

1. We say $X$ is connected if $X$ is not empty and whenever $X = T_1 \amalg T_2$ with $T_ i \subset X$ open and closed, then either $T_1 = \emptyset$ or $T_2 = \emptyset$.

2. We say $T \subset X$ is a connected component of $X$ if $T$ is a maximal connected subset of $X$.

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