Continuous functions premap closed sets to closed sets

up:: Topologically Continuous Function

Let $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ be a continuous function.

Then we know that

$$\forall U_Y\in {\tau }_Y,f^{-1}(U_Y)\in {\tau }_X⟺\forall U_Y\in {\tau }_Y,f^{-1}(U_Y)=U_X\in {\tau }_X$$

Let $\mathcal{F}(\tau )$ be the set of closed sets associated to the topology $\tau$. Since The preimage of the complement is the complement of the preimage, we have that (using $F_Y=Y\setminus U_Y\in \mathcal{F}({\tau }_Y)$, and $F_X=X\setminus U_X\in \mathcal{F}({\tau }_X)$)

$$f^{-1}(F_Y)=f^{-1}(Y\setminus U_Y)=f^{-1}(Y)\setminus f^{-1}(U_Y)=X\setminus U_X\in \mathcal{F}({\tau }_X)$$

Thus, continuous functions premap closed sets to closed sets.

Corollaries

This definition is also equivalent to The closure of a continuous image of a closure is the closure of the image. That is, for all subsets $D\in \mathbb{P}(X)$, we have that $f(\overline{D})=\overline{f(D)}$.