Let $X$ be a topological space and $x _ { 0 } \in X$. Let $\mathcal { S } = \left\{ B \subseteq X \mid x _ { 0 } \in B \text{ and } B \text{ is connected} \right\}$. Let $$A = \bigcup _ { B \in \mathcal { S } } B .$$ Show that $A$ is closed.
Let $X$ be a topological space and $x _ { 0 } \in X$. Let $\mathcal { S } = \left\{ B \subseteq X \mid x _ { 0 } \in B \text{ and } B \text{ is connected} \right\}$. Let
$$A = \bigcup _ { B \in \mathcal { S } } B .$$
Show that $A$ is closed.