Perfect compactification
A compactification of a completely-regular space
such that the closure in
of the boundary of any open set
coincides with the boundary of
, where
is the maximal open set in
for which
. Equivalent definitions are as follows:
a) for any pair of disjoint open sets
;
b) if a closed set partitions
into open sets
and
, then the closure of
in
partitions
into
and
;
c) does not partition
locally at any of its points.
A compactification is perfect if and only if the natural mapping
is monotone; here
is the Stone–Čech compactification of
. Also,
is the unique perfect compactification of
if and only if
with
a compactum and
. The local connectedness of
implies the local connectedness of any perfect extension
of
satisfying the first axiom of countability (and also the local connectedness of all intermediate extensions). Among all the perfect compactifications of
there is a minimal one,
, if and only if
has at least one compactification with punctiform remainder (cf. Remainder of a space). The remainder in
is punctiform and
is the maximal such extension among those with punctiform remainder. Every homeomorphism of
extends to a homeomorphism of
, and every perfect mapping from
onto
extends to a mapping from
onto
(provided
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) |
Perfect compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perfect_compactification&oldid=19245