Continuous function between topological spaces

For \(f:\mathbb{R}\to\mathbb{R}\) with the usual topology, the definition agrees with the \(\varepsilon\)–\(\delta\) one: the preimage of every open interval contains an interval around each of its points. The advantage is that it applies to any pair of spaces, even without metrics, and is handled with a single idea: preimages.

Preimages respect unions and intersections, \(f^{-1}(\bigcup V_i)=\bigcup f^{-1}(V_i)\) and \(f^{-1}(V\cap W)=f^{-1}(V)\cap f^{-1}(W)\). Hence it suffices to check the condition on the open sets of a base of \(Y\). Also, the composition of continuous maps is continuous, since \((g\circ f)^{-1}(W)=f^{-1}(g^{-1}(W))\).

Continuity depends on both topologies. If the domain is discrete, every function is continuous; if the target is trivial, every function is continuous; constants always are. By contrast, the identity of a set, from one topology to another, is continuous exactly when the domain’s topology contains all the open sets of the target’s (is finer).

Typical mistake: confusing continuity with sending open sets to open sets. A continuous function may send an open set to a non-open set (the constant \(x\mapsto0\) sends \(\mathbb{R}\) to \(\{0\}\)), and a function sending open sets to open sets need not be continuous. If it is also bijective and its inverse is continuous too, we have a homeomorphism.