Continuous functions premap closed sets to closed sets

up:: Topologically Continuous Function

Let be a continuous function.

Then we know that

Let be the set of closed sets associated to the topology . Since The preimage of the complement is the complement of the preimage, we have that (using , and )

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 , we have that .