nLab
continuous map

A continuous map is a morphism in the category of topological spaces and in some similar contexts like locales and convergence spaces.

A function $f:X\to Y$ between topological spaces is a continuous map if for every open subset $U\subset Y$, the preimage $f^{-1}(U)$ is open in $X$. Continuous maps of topological spaces preserve connectedness and send compact subsets to compact subsets. However it is not necessary that the preimage of a compact set be compact; the continuous maps such that the preimage of any compact set is compact are called proper maps. A continuous map of topological spaces which is invertible as a function of sets, is a homeomorphism if the inverse function? is a continuous map as well. A continuous map is open (resp. closed) if it sends open (resp. closed) subsets to open (closed) subsets. Every continuous map from a compact space to a Hausdorff space is both closed and proper.

A function $f$ between convergence spaces is continuous if for any filter $F$ such that $F\to x$, it follows that $f(F)\to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A)\mid A\in F\}$.

A continuous map between locales is simply a frame homomorphism in the opposite direction.