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Book: Kechris, Descriptive Set Theory

Theorem (Pg 75): If $X$ is Polish, the Effros Borel space of $F(X)$ is standard.

He begins by letting $\overline{X}$ be a compactification of $X$. Then Consider the map sending $F \in F(X)$ to $\overline{F} \in K(\overline{X})$, where $\overline{F}$ denotes the closure of $F$ in the $\overline{X}$. This map is 1-1. Also it is proved that $G=\{\overline{F}: F \in F(X)\}$ is a $G_{\delta}$ in $K(\overline{X})$. Thus $G$ is Polish.

Fine. But then the book states, "Transfer back to $F(X)$ its topology via the bijection $F \rightarrow \overline{F}$ to get a Polish topology $\tau$ on $F(X)$."

I've never heard of transferring a topology via a map, or bijection. What exactly does he mean?

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If $f: X \to Y$ is a bijection between two sets, and $Y$ has a topology, there is a unique topology on $X$ that makes $f$ a homeomorphism; namely, declare $U$ to be open if and only if $f(U)$ is open. This is what they mean.