Prove that if $W$ is an open subset of $Y$ then the set $\{x\in X: f(x)\in W\}$ is an open subset of $X$.

Question

Suppose that $f:X\longrightarrow Y$ is continuous. Prove that if $W$ is an open subset of $Y$ then the set $\{x\in X: f(x)\in W\}$ is an open subset of $X$.

$\textbf{Proof:}$ Assume $f:X\longrightarrow Y$ is continuous and let $W\subset Y$ be open.

Consider the set $G=\{x\in X: f(x)\in W\}$.

If $G=\emptyset$ then $G$ is clearly open in $X$ and we are done.

Otherwise let $G\neq\emptyset$ and consider $x\in G$.

Since $f:X\longrightarrow Y$ is continuous and $f(x)\in W$, $W$ is open.

Thus $f^{-1}(W)$ is open in $X$.

Hence $G=f^{-1}(W)\in X$.

Then $G\subset X$.

Thus $G$ is an open subset in $X$.

Therefore if $f:X\longrightarrow Y$ is continuous and $W$ is an open subset of $Y$, then the set $\{x\in X: f(x)\in W$ is an open subset of $X$.

Just wondering if I did this right or if I can improve on something. Please let me know of any mistakes or clarifications that I need to make. Thanks.

I would say that they are obviously topological spaces and, since the problem deals with continuity, I would not consider it necessary to say that. — user247327, May 11 '18 at 18:10
But if $X$ and $Y$ are topological spaces this question is pointless, what he asks $is$ the definition of continuity doesn't it? — 3nondatur, May 11 '18 at 18:16
@user247327 But the common definition of continuity in topological spaces is exactly "the preimage of open set is open"... — Alex Vong, May 11 '18 at 18:17
@user247327 I think $X$ and $Y$ are assumed to be metric spaces so that we can use the $\varepsilon$-$\delta$ definition instead. — Alex Vong, May 11 '18 at 18:31

score 1 · Answer 1 · answered May 11 '18 at 18:20

1

No, it is not right:

You wrote that “Since $f:X\longrightarrow Y$ is continuous and $f(x)\in W$, $W$ is open.” No: $W$ is open because you are assuming that it is open.
“Thus $f^{-1}(W)$ is open in $X$.” That's what you are supposed to prove!

answered May 11 '18 at 18:20

José Carlos Santos

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score 0 · Accepted Answer · answered May 11 '18 at 18:16

It's not correct. $G=f^{-1}(W)$ so when you say "Thus $f^{-1}(W)$ is open in $X$", you didn't give any proof of this assertion! Moreover, this assertion is exactly... what you're asked to prove!

To prove it, you need to apply the definitions of continuity and openness. It depends on the spaces that you consider: as said in the comments above, if you use general topological spaces, then it follows directly from the definitions. Otherwise, you must tell us if you deal with metric spaces or...

Prove that if $W$ is an open subset of $Y$ then the set $\{x\in X: f(x)\in W\}$ is an open subset of $X$.

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