Section 16: Problem 1 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

Show that if $Y$ is a subspace of $X$ , and $A$ is a subset of $Y$ , then the topology $A$ inherits as a subspace of $Y$ is the same as the topology it inherits as a subspace of $X$ .

$B$ is open in $A\subset Y$ iff $B=A\cap C$ and $C$ is open in $Y\subset X$ iff $B=A\cap C$ , $C=Y\cap D$ and $D$ is open in $X$ iff $B=A\cap Y\cap D$ and $D$ is open in $X$ iff $B=A\cap D$ and $D$ is open in $X$ iff $B$ is open in $A\subset X$ .

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Section 16: Problem 1 Solution

Section 16: Problem 1 Solution