continuous maps

Continuous Maps

Roughly, a continuous map is a mapping that is open preserving, i.e., a function \(f: X\rightarrow Y\) where \(X, Y\) are top spaces, such that the preimage of any open in \(Y\) is open in \(X\). In other words: \(\forall O\in\mathcal{O}(Y): f^{-1}(O) \in \mathcal{O}(X). \) The identity map \(\operatorname{id}_X\) is continuous for any top space \(X\).

In order for \(f\) to be continuous, it is enough that the preimages of sets comprising some prebase of \(Y\) be open.

Composition

If \( f: X\to Y\) and \(g: Y \to Z\) are continuous then \(g\circ f: X \to Z\) also is continuous. Indeed, for all open \(O \in \mathcal{O}(Z)\), \((g\circ f)^{-1}(O) = f^{-1}(g^{-1}(O)) \). But \(g^{-1}(O)\) is open in \(Y\) since \(g\) is continuous. Then \(f^{-1}(g^{-1}(O))\) is open in \(X\).

Modifications continuous

Let \(f: X\to Y\) be continous. For any \(A \subset X\) and \(B\subset Y\), if \(f(A) \subset B\), then the compression \(\operatorname{ab} f: A\to B\) is continuous. In particular, the restriction \(f\vert_A\) is continuous. Similarly, \(\operatorname{ab} f: X \to B\) is continous if and only if \(f: X\to Y\) is continuous. In particular, \(f: X\to Y\) is continuous if and only if \(f: X\to f(X)\) is continuous.

Continuity and fundamental covers

If \(\Gamma\) is a fundamental cover of \(X\), then \(f:X\to Y\) is continuous if and only if \( f\vert_{A'} : A' \to Y\) is continuous for any \(A' \in \Gamma.\)