= Perfect-map characterization by Stone-Čech remainders
{title2=$f\text{ perfect}\iff(\beta f)^{-1}(Y)=X$}
For a <continuous map> $f:X\to Y$ between <Tychonoff spaces>, extend it to $F:\beta X\to\beta Y$ by the <Stone-Čech extension property>. If $f$ is perfect, <compactness> of a <fiber> and closedness of $f$ show that no remainder point can map to a point of $Y$: separate such a point from the <fiber> by a neighborhood in $\beta X$, and use closedness to obtain a target neighborhood whose preimage stays inside it, contradicting density. Conversely, if $F^{-1}(Y)=X$, each <fiber> is <compact>, and for closed $A\subseteq X$ the equality $f(A)=F(\overline A^{\beta X})\cap Y$ proves closedness. Equivalently $F(\beta X\setminus X)\subseteq\beta Y\setminus Y$.
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