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A [[topological space]] $X$ is $0$-dimensional if it is a $T_1$-space (cf. also [[Separation axiom|Separation axiom]]) with a base of [[clopen]] sets (a set is called clopen if it is both open and closed). The Banaschewski compactification [[#References|[a1]]], [[#References|[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.
 
A [[topological space]] $X$ is $0$-dimensional if it is a $T_1$-space (cf. also [[Separation axiom|Separation axiom]]) with a base of [[clopen]] sets (a set is called clopen if it is both open and closed). The Banaschewski compactification [[#References|[a1]]], [[#References|[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]] [[#References|[a4]]] (as generalized by N.A. Shanin, [[#References|[a3]]]). A fairly general approach subsuming the above-mentioned compactifications is as follows.
 
The Banaschewski compactification is also a special case of the [[Wallman compactification]] [[#References|[a4]]] (as generalized by N.A. Shanin, [[#References|[a3]]]). A fairly general approach subsuming the above-mentioned compactifications is as follows.
  
Let $X$ be an arbitrary non-empty set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b1101209.png" /> a [[Lattice|lattice]] of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012011.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012012.png" /> is disjunctive and separating, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012013.png" /> be the [[Algebra|algebra]] generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012014.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012015.png" /> be the set of non-trivial zero-one valued finitely additive measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012016.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012017.png" /> be the set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012018.png" /> that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012019.png" />-regular, i.e.,
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Let $X$ be an arbitrary non-empty set and $\mathcal{L}$ a [[Lattice|lattice]] of subsets of $X$ such that $\emptyset$. Assume that $\mathcal{L}$ is disjunctive and separating, let $\mathcal{A}(\mathcal{L})$ be the [[Algebra|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.,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012020.png" /></td> </tr></table>
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$$
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\mu(A) = \sup \{ \mu(L) : L\subset A, L \in \mathcal{L}\}, \quad A \in \mathcal{A}(\mathcal{L}).
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$$
  
One can identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012021.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012022.png" />-prime filters and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012023.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012024.png" />-ultrafilters (cf. also [[Filter|Filter]]; [[Ultrafilter|Ultrafilter]]).
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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|Filter]]; [[Ultrafilter|Ultrafilter]]).
  
Next, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012026.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012027.png" /> is a lattice isomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012028.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012029.png" />. Take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012030.png" /> as a base for the closed sets of a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012032.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012033.png" /> is a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012034.png" />-space and it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012035.png" /> (cf. [[Hausdorff space|Hausdorff space]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012036.png" /> is a normal lattice. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012037.png" /> can be densely imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012038.png" /> by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012040.png" /> is the Dirac measure concentrated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012041.png" /> (cf. also [[Dirac delta-function|Dirac delta-function]]). The mapping is a [[Homeomorphism|homeomorphism]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012042.png" /> is given the topology of closed sets with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012043.png" /> as base for the closed sets.
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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|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|Dirac delta-function]]). The mapping is a [[Homeomorphism|homeomorphism]] if $X$ is given the topology of closed sets with $\mathcal{L}$ as base for the closed sets.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012044.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012045.png" />-space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012046.png" /> is the lattice of closed sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012047.png" /> becomes the usual Wallman compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012048.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012049.png" /> is a Tikhonov space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012050.png" /> is the lattice of zero sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012051.png" /> becomes the Stone–Čech compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012052.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012053.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012054.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012055.png" />-space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012056.png" /> is the lattice of clopen sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012057.png" /> becomes the Banaschewski compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012058.png" />.
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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 $\beta_0 X$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012059.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012060.png" /> is a [[Normal space|normal space]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012061.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012062.png" /> is strongly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110120/b11012064.png" />-dimensional (i.e., the clopen sets separate the zero sets).
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$\omega X = \beta X$ if and only if $X$ is a [[Normal space|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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Banaschewski,  "Über nulldimensional Räume"  ''Math. Nachr.'' , '''13'''  (1955)  pp. 129–140</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Banaschewski,  "On Wallman's method of compactification"  ''Math. Nachr.'' , '''27'''  (1963)  pp. 105–114</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.A. Shanin,  "On the theory of bicompact extensions of topological spaces"  ''Dokl. Aka. Nauk SSSR'' , '''38'''  (1943)  pp. 154–156  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Wallman,  "Lattices and topological spaces"  ''Ann. Math.'' , '''39'''  (1938)  pp. 112–126</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Banaschewski,  "Über nulldimensional Räume"  ''Math. Nachr.'' , '''13'''  (1955)  pp. 129–140</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Banaschewski,  "On Wallman's method of compactification"  ''Math. Nachr.'' , '''27'''  (1963)  pp. 105–114</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.A. Shanin,  "On the theory of bicompact extensions of topological spaces"  ''Dokl. Aka. Nauk SSSR'' , '''38'''  (1943)  pp. 154–156  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Wallman,  "Lattices and topological spaces"  ''Ann. Math.'' , '''39'''  (1938)  pp. 112–126</TD></TR></table>
  
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Latest revision as of 02:22, 15 February 2024

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
How to Cite This Entry:
Banaschewski compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banaschewski_compactification&oldid=54784
This article was adapted from an original article by G. Bachman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article