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Is the image of a $G_\delta$ set under a continuous injection again a $G_\delta$? [closed]

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Question: Is the image of a $G_\delta$ set in $X$ under some continuous injection $f:X\to Y$ again a $G_\delta$ set in $Y$?

I want to know this mainly for the case where $X=[0,1]$ and $Y=\Bbb R$. But, I also want to know if this is true in general?

A $G_\delta$ set is a set which can be expressed as a countable intersection of open sets.

My attempt: I think yes. I know a $G_\delta$ set is a set which can be expressed as a countable intersection of open sets. For proving the given statement, let $S$ be any $G_\delta$ subset of $[0,1]$. Then we need to prove that $f(S)$ is also a $G_\delta$ set. That is, we need to show that $f(S)$ can be expressed as a countable intersection of open sets. I am unable to proceed further from this.


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