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 $\mathcal{L}$ a lattice of subsets of $X$ such that $\emptyset$. Assume that $\mathcal{L}$ is disjunctive and separating, let $\mathcal{A}(\mathcal{L})$ be the algebra generated by $\mathcal{L}$, let $\mathcal{A}(\mathcal{L})$ be the set of non-trivial zero-one valued finitely additive measures on $\mathcal{A}(\mathcal{L})$, and let $I_R(\mathcal{L})$ be the set of elements $\mu\in I(\mathcal{L})$ that are $\mathcal{L}$-regular, i.e.,

$$ \mu(A) = \sup \{ \mu(L) : L\subset A, L \in \mathcal{L}\}, \quad A \in \mathcal{A}(\mathcal{L}). $$

One can identify $I(\mathcal{L})$ with the $\mathcal{L}$-prime filters and $I_R(\mathcal{L})$ with the $\mathcal{L}$-ultrafilters (cf. also Filter; Ultrafilter).

Next, let $V(A) = \{\mu \in I_R(\mathcal{L}) : \mu(A) = 1\}$, where $A \in \mathcal{A}(\mathcal{L})$; $V$ is a lattice isomorphism from $\mathcal{L}$ to $V(\mathcal{L} = \{ V(L) : L \in \mathcal{L}\}$. Take $V(\mathcal{L})$ as a base for the closed sets of a topology $\tau V(\mathcal{L})$ on $I_R(\mathcal{L})$. Then $(I_R(\mathcal{L}), \tau V(\mathcal{L}))$ is a compact $T_1$-space and it is $T_2$ (cf. Hausdorff space) if and only if $\mathcal{L}$ is a normal lattice. $X$ can be densely imbedded in $I_R(\mathcal{L})$ by the mapping $x \to \mu_x$, where $\mu$ is the Dirac measure concentrated at $x$ (cf. also Dirac delta-function). The mapping is a homeomorphism if $X$ is given the topology of closed sets with $\mathcal{L}$ as base for the closed sets.

If $X$ is a $T_1$-space and $\mathcal{L}$ is the lattice of closed sets, then $(I_R(\mathcal{L}), \tau V(\mathcal{L}))$ becomes the usual Wallman compactification $\omega X$.

If $X$ is a Tikhonov space and $\mathcal{L}$ is the lattice of zero sets, then $(I_R(\mathcal{L}), \tau V(\mathcal{L}))$ becomes the Stone–Čech compactification $\beta X$.

If $X$ is a $0$-dimensional $T_1$-space and $\mathcal{L}$ is the lattice of clopen sets, then $(I_R(\mathcal{L}), \tau V(\mathcal{L}))$ becomes the Banaschewski compactification $\beta_0 X$.

$\omega X = \beta X$ if and only if $X$ is a normal space; $\beta X = \beta_0 X$ if and only if $X$ is strongly $0$-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