Intersection of Closed Sets is Closed

Last updated Nov 1, 2022

Suppose $(X, \tau)$ is a Topological Space. Suppose $\mathcal{K} \subset \mathcal{P}(X)$ is a collection of closed sets in $X$. Then $\bigcap\limits \mathcal{K}$ is Closed in $X$.

This follows from De Morgan’s Law. Index $\mathcal{K}$ with $I$. That is, ${K_{\alpha}}{\alpha \in I} = \mathcal{K}$. Then $K{\alpha}^{C}$ is Open in $X$ since $K_{\alpha}$ is Closed for all $\alpha \in I$. Thus,

$$\begin{align*} \Big( \bigcap\limits_{\alpha \in I} K_{\alpha} \Big)^{C} &= \bigcup\limits_{\alpha \in I} K_{\alpha}^{C} & \text{De Morgan’s Law}\\ & \in \tau \end{align*}$$

since $\tau$ is closed under unions. Thus $\bigcap\limits_{\alpha \in I} K_\alpha$ is Closed. $\blacksquare$