Showing $q:A\to p(A)$ is an open map.

Question

I'm working on this problem out of Munkres: Let $p:X\to Y$ be an open map. Show if $A$ is open in $X$, then the map $q:A\to p(A)$ obtained by restricting $p$ is an open map. $\newcommand{\uset}{\mathcal{U}}$ $\newcommand{\oset}{\mathcal{O}}$

This seems like a very straightforward problem. Let $U$ be an open set in $A$. Then $U=\oset\cap A$ for some open set $\oset$ of $X$. Then we would like to have $$q(U)=p(\oset\cap A)=p(\oset)\cap p(A)$$

which would be open in $p(A)$ because $p(\oset)$ is an open set in $Y$. However, they don't give me any more information about the map (like injectivity) so all I can guarantee is that $$q(U)=p(\oset\cap A)\subseteq p(\oset)\cap p(A)$$

and I'm not sure how I can be sure $q(U)$ is open. Is there something I'm overlooking?

I assume here you intend $q$ to be an open map with respect to the subspace topology? — charlestoncrabb, Nov 05 '15 at 00:06
If $A$ is open in $X$ and $\mathcal{O}$ is open in $X$, then $A\cap \mathcal{O}$ is open in $X$. — J126, Nov 05 '15 at 00:07
Since $A$ is open in $X$ you're allowed to assume that $U=\mathcal O$. — Arthur, Nov 05 '15 at 00:07

score 5 · Accepted Answer · answered Nov 05 '15 at 00:11

5

Any open set $U$ in $A$ is also open in $X$. Then $q(U)=p(U)$ is open in $Y$ so it is also open in $p(A)$ since open set in $p(A)$ are just open set in $Y$ contained in $p(A)$.

answered Nov 05 '15 at 00:11

Ben

3,798

Showing $q:A\to p(A)$ is an open map.

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