View source for Stone space

''of a [[Boolean algebra]] $\mathcal{B}$''

A totally-disconnected compact space $(X,\mathfrak{T})$ whose field of all open-and-closed sets is isomorphic to $\mathcal{B}$. This space is defined canonically from $\mathcal{B}$ in the following way: $B$ is the set of all [[ultrafilter]]s of $\mathcal{B}$, while the topology $\mathfrak{T}$ is generated by the family of subsets of the form $U_A = \{ \xi \in X : A \in \xi \}$, where $A$ is an arbitrary element of $\mathcal{B}$. Instead of ultrafilters, the set of maximal ideals, of two-valued homomorphisms or of two-valued measures on $\mathcal{B}$ with an appropriate topology may be used. Isomorphic Boolean algebras have homeomorphic Stone spaces. Every totally-disconnected compact space is the Stone space of the Boolean algebra of its open-and-closed sets.

The concept of a Stone space and its basic properties were discovered and studied by M.H. Stone (1934–1937, see [[#References|[1]]]).

The Stone space of a Boolean algebra is metrizable if and only if the Boolean algebra is countable. A Boolean algebra is complete if and only if its Stone space is extremally disconnected (i.e. if the closure of any open set in the space is open). The perfect [[Cantor set]] is the Stone space of a countable atomless Boolean algebra (they are all isomorphic). The generalized Cantor discontinuum $D^m$ is the Stone space of the free Boolean algebra with $m$ generators.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  R. Sikorski,   "Boolean algebras" , Springer  (1969)</TD></TR>
</table>



====Comments====
An ultrafilter on a Boolean algebra is a maximal [[filter]] on the underlying ordered set.

A good up-to-date reference on Boolean algebras is [[#References|[a2]]].

The term  "Stone space"  is often used as a synonym for  "totally-disconnected compact space" , although others use  "Boolean space"  for this concept.  A topological space is a Stone space if and only if it is compact, Hausdorff and totally disconnected; compact and [[Totally separated space|totally separated]]; compact, [[T0 space|$T_0$]] and [[Zero-dimensional space|zero-dimensional]]; or Hausdorff and coherent.

The correspondence between Boolean algebras and their Stone spaces is a categorical duality (see [[Dual category]]); thus, if $\mathcal{B}$ and $\mathcal{B}_1$ are Boolean algebras with Stone spaces $X$ and $X_1$ respectively, the Boolean homomorphisms $\mathcal{B} \rightarrow \mathcal{B}_1$ correspond bijectively to continuous functions $X_1 \rightarrow X$. Using this, one may translate algebraic theorems about Boolean algebras into topological theorems about Stone spaces, or vice versa: for example, Sikorski's theorem that complete Boolean algebras are injective in the category of all Boolean algebras (see [[Injective object]]) corresponds to Gleason's theorem that extremally-disconnected spaces are projective in the category of Stone spaces. For more details see [[#References|[a1]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.T. Johnstone,   "Stone spaces" , Cambridge Univ. Press  (1982) ISBN 0-521-33779-8</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Koppelberg,   "General theory of Boolean algebras"  J.D. Monk (ed.)  R. Bonnet (ed.) , ''Handbook of Boolean algebras'' , '''3''' , North-Holland  (1989)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.D. Monk (ed.)  R. Bonnet (ed.) , ''Handbook of Boolean algebras'' , '''1–3''' , North-Holland  (1989)</TD></TR>
</table>

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