Banaschewski compactification

A topological space $X$ is $0$-dimensional if it is a $T_1$-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 $X$, denoted by $\beta_0 X$, is the $0$-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 $X$ 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.,

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