Perfect compactification

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).

Comments

A space is called punctiform if and only if no compact connected subset contains more than one point.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 232ff (Translated from Russian)