restriction of a continuous mapping is continuous
Theorem Suppose and are topological spaces, and suppose
is a continuous function
. For a subset ,
the restriction
(http://planetmath.org/RestrictionOfAFunction)
of to (that is ) is a continuous
mapping , where is given the subspace topology
from .
Proof. We need to show that for any open set , we can write for some set that is open in . However, by the properties of the inverse image (see this page (http://planetmath.org/InverseImage)), we have for any open set ,
Since is continuous, is open in , and our claim follows.
Title | restriction of a continuous mapping is continuous |
---|---|
Canonical name | RestrictionOfAContinuousMappingIsContinuous |
Date of creation | 2013-03-22 13:55:53 |
Last modified on | 2013-03-22 13:55:53 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 6 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 54C05 |
Classification | msc 26A15 |