Section 22*: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $p:X\rightarrow Y$ be an open map. Show that if $A$ is open in $X$ , then the map $q:A\rightarrow p(A)$ obtained by restricting $p$ is an open map.

If $U\subset A$ is open in $A$ , then, since $A$ is open in $X$ , $U$ is open in $X$ , hence, $q(U)=p(U)\subset p(A)$ is open in $X$ , and in $p(A)$ .

This exercise, I believe, shows, in particular, that, in Theorem 22.1, if $p$ is an open quotient map, and $A$ is open, then even if $A$ is not saturated, the restriction $q$ is an open quotient map. In other words, in Theorem 22.1, instead of assuming that $A$ is saturated, we can assume that both (1) $A$ is open and (2) $p$ is open.

Subpages

Section 22*: Problem 5 Solution

Section 22*: Problem 5 Solution