Banaschewski compactification
A topological space is
-dimensional if it is a
-space (cf. also Separation axiom) with a base of clopen sets (a set is called clopen if it is both open and closed). The Banaschewski compactification [a1], [a2] of
, denoted by
, is the
-dimensional analogue of the Stone–Čech compactification of a Tikhonov space. It can be obtained as the Stone space of the Boolean algebra of clopen subsets.
The Banaschewski compactification is also a special case of the Wallman compactification [a4] (as generalized by N.A. Shanin, [a3]). A fairly general approach subsuming the above-mentioned compactifications is as follows.
Let be an arbitrary non-empty set and
a lattice of subsets of
such that
. Assume that
is disjunctive and separating, let
be the algebra generated by
, let
be the set of non-trivial zero-one valued finitely additive measures on
, and let
be the set of elements
that are
-regular, i.e.,
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One can identify with the
-prime filters and
with the
-ultrafilters (cf. also Filter; Ultrafilter).
Next, let , where
;
is a lattice isomorphism from
to
. Take
as a base for the closed sets of a topology
on
. Then
is a compact
-space and it is
(cf. Hausdorff space) if and only if
is a normal lattice.
can be densely imbedded in
by the mapping
, where
is the Dirac measure concentrated at
(cf. also Dirac delta-function). The mapping is a homeomorphism if
is given the topology of closed sets with
as base for the closed sets.
If is a
-space and
is the lattice of closed sets, then
becomes the usual Wallman compactification
.
If is a Tikhonov space and
is the lattice of zero sets, then
becomes the Stone–Čech compactification
.
If is a
-dimensional
-space and
is the lattice of clopen sets, then
becomes the Banaschewski compactification
.
if and only if
is a normal space;
if and only if
is strongly
-dimensional (i.e., the clopen sets separate the zero sets).
References
[a1] | B. Banaschewski, "Über nulldimensional Räume" Math. Nachr. , 13 (1955) pp. 129–140 |
[a2] | B. Banaschewski, "On Wallman's method of compactification" Math. Nachr. , 27 (1963) pp. 105–114 |
[a3] | N.A. Shanin, "On the theory of bicompact extensions of topological spaces" Dokl. Aka. Nauk SSSR , 38 (1943) pp. 154–156 (In Russian) |
[a4] | H. Wallman, "Lattices and topological spaces" Ann. Math. , 39 (1938) pp. 112–126 |
Banaschewski compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banaschewski_compactification&oldid=15460