TITLE: Complement of a Topology
QUESTION [7 upvotes]: Let $(X, \tau)$ topology, I was wondering, if given $$\tau'=\{A^C \mid A \in \tau\}$$ did $\tau'$ also a topology on $X$? If so, why? Thank you.

REPLY [3 votes]: Example 1: Let $\Bbb R$ with usual topology. $\mathscr T=\{(a,b): a,b \in \Bbb R \}$ is the topology of $\Bbb R$. However $\mathscr T'$ is not the topology of $\Bbb R$. Since $\{[\frac1n,\infty]: n\in \Bbb N^+\} $ and $\{[-\infty, -\frac{1}n]: n\in \Bbb N^+\} $ are the elements of $\mathscr T'$. However their union is $\Bbb R\setminus \{0\}$, which is not the element of $\mathscr T'$. Therefore $\mathscr T'$ is not a topology on $\Bbb R$.