# Perfect compactification

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A compactification $Y$ of a completely-regular space $X$ such that the closure in $Y$ of the boundary of any open set $U \subset X$ coincides with the boundary of $O( U)$, where $O( U)$ is the maximal open set in $Y$ for which $O( U) \cap X = U$. Equivalent definitions are as follows:

a) $O( U \cup V )= O( U) \cup O( V)$ for any pair of disjoint open sets $U, V$;

b) if a closed set $F$ partitions $X$ into open sets $U$ and $V$, then the closure of $F$ in $Y$ partitions $Y$ into $O( U)$ and $O( V)$;

c) $Y \setminus X$ does not partition $Y$ locally at any of its points.

A compactification $\gamma X$ is perfect if and only if the natural mapping $\beta \mathop{\rm id} _ {X} : \beta X \rightarrow \gamma X$ is monotone; here $\beta$ is the Stone–Čech compactification of $X$. Also, $\beta X$ is the unique perfect compactification of $X$ if and only if $X= A \cup M$ with $A$ a compactum and $\mathop{\rm dim} M = 0$. The local connectedness of $X$ implies the local connectedness of any perfect extension $Y$ of $X$ satisfying the first axiom of countability (and also the local connectedness of all intermediate extensions). Among all the perfect compactifications of $X$ there is a minimal one, $\mu X$, if and only if $X$ has at least one compactification with punctiform remainder (cf. Remainder of a space). The remainder in $\mu X$ is punctiform and $\mu X$ is the maximal such extension among those with punctiform remainder. Every homeomorphism of $X$ extends to a homeomorphism of $\mu X$, and every perfect mapping from $X$ onto $X ^ \prime$ extends to a mapping from $\mu X$ onto $\mu X ^ \prime$( provided $\mu X ^ \prime$ exists).