Complements, Unions and Intersections of Closed Sets in a Topological Space

Let $(X,\mathcal{T})$ be a topological space. For a subset $S\subseteq X$ write $X\setminus S$ for the complement of $S$ relative to $X$, so that $S$ is closed in $X$ exactly when $X\setminus S\in\mathcal{T}$. Let $\mathbb{N}$ denote the natural numbers.

Then the following hold.

1. The empty set $\varnothing$ and the whole set $X$ are closed in $X$.

2. For every $n\in\mathbb{N}$ and all subsets $C_1,\dots,C_n\subseteq X$ that are closed in $X$, the set of all $x\in X$ such that $x\in C_i$ for at least one $i\in\{1,\dots,n\}$ is closed in $X$.

3. For every set $I$ and every family of subsets of $X$ $(C_a)_{a\in I}$ such that $C_a$ is closed in $X$ for every $a\in I$, the set of all $x\in X$ such that $x\in C_a$ for every $a\in I$ is closed in $X$.