restriction of a continuous mapping is continuous

Theorem Suppose $X$ and $Y$ are topological spaces, and suppose $f:X\to Y$ is a continuous function. For a subset $A\subset X$, the restriction (http://planetmath.org/RestrictionOfAFunction) of $f$ to $A$ (that is ${f|}_A$) is a continuous mapping ${f|}_A:A\to Y$, where $A$ is given the subspace topology from $X$.

Proof. We need to show that for any open set $V\subset Y$, we can write ${({f|}_A)}^{-1}(V)=A\cap U$ for some set $U$ that is open in $X$. However, by the properties of the inverse image (see this page (http://planetmath.org/InverseImage)), we have for any open set $V\subset Y$,

$${({f|}_A)}^{-1}(V)=A\cap f^{-1}(V).$$

Since $f:X\to Y$ is continuous, $f^{-1}(V)$ is open in $X$, and our claim follows. $\mathrm{\square }$

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