Continuous maps that are not closed or not open

We recall some definitions on open and closed maps. In topology an open map is a function between two topological spaces which maps open sets to open sets. Likewise, a closed map is a function which maps closed sets to closed sets.

For a continuous function $$f: X \mapsto Y$$, the preimage $$f^{-1}(V)$$ of every open set $$V \subseteq Y$$ is an open set which is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in $$Y$$ are closed in $$X$$. However, a continuous function might not be an open map or a closed map as we prove in following counterexamples. Continue reading Continuous maps that are not closed or not open