Remark 5.7.4. Let $X$ be a topological space and $x \in X$. Let $Z \subset X$ be the connected component of $X$ passing through $x$. Consider the intersection $E$ of all open and closed subsets of $X$ containing $x$. It is clear that $Z \subset E$. In general $Z \not= E$. For example, let $X = \{ x, y, z_1, z_2, \ldots \} $ with the topology with the following basis of opens, $\{ z_ n\} $, $\{ x, z_ n, z_{n + 1}, \ldots \} $, and $\{ y, z_ n, z_{n + 1}, \ldots \} $ for all $n$. Then $Z = \{ x\} $ and $E = \{ x, y\} $. We omit the details.

[Example 6.1.24, Engelking]

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