composition of continuous mappings is continuous

Theorem 1.

The composition of two continuous mappings (when defined) is continuous.

Proof.

Let $X,Y,Z$ be topological space, and let $f,g$ be mappings

$f:X$ $\to$ $Y,$

$g:Y$ $\to$ $Z.$

We wish to prove that $g\circ f$ is continuous. Suppose $B$ is an open set in $Z$. Since $g$ is continuous, $g^{-1}(B)$ is an open set in $Y$, and since $f$ is continuous, $f^{-1}(g^{-1}(B))$ is an open set in $X$. Since $f^{-1}(g^{-1}(B))={(g\circ f)}^{-1}(B)$, it follows that ${(g\circ f)}^{-1}(B)$ is open and the composition if continuous. ∎

Title	composition of continuous mappings is continuous
Canonical name	CompositionOfContinuousMappingsIsContinuous
Date of creation	2013-03-22 15:16:52
Last modified on	2013-03-22 15:16:52
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 26A15
Classification	msc 54C05