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|
|Date of creation||2013-03-22 13:55:53|
|Last modified on||2013-03-22 13:55:53|
|Last modified by||matte (1858)|