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I know the theorem ---

Let $(X, d)$ and $(X', d')$ be metric spaces. $f: X \mapsto X'$ is a continuous function if and only if for every open set $U \subset X'$, $f^{-1}(U)\subset X$ is open in X.

Does that mean for every open set $A \subset X$, $f(A) \subset X'$ is open? Basically, I'm asking whether the part that comes after "if and only if" works conversely?

user1691278
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1 Answers1

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No. The maps with the property that $f(A)\subset X'$ is open whenever $A\subset X$ is open are called open maps. Not all open maps are continuous. See here: Open maps which are not continuous

Conversely, each constant map is continuous, but clearly not open.

ervx
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