composition of continuous mappings is continuous
Theorem 1.
The composition of two continuous mappings (when
defined) is continuous
.
Proof.
Let be topological space, and let be
mappings
We wish to prove that is continuous. Suppose is an open set in . Since is continuous, is an open set in , and since is continuous, is an open set in . Since , it follows that 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 |