boolean_spaces - MathStructures

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=====Boolean spaces=====

Abbreviation: **BSp**

====Definition====
A \emph{Boolean space} is a [[compact Hausdorff topological space]] $\mathbf{X}=\langle X,\Omega\rangle$ that is \emph{totally disconnected}:

any two distinct points are separated by a clopen set ($\forall x\ne y\in X\exists U\in\Omega (x\in X\text{ and }y\in X\setminus U\in\Omega)$).

==Morphisms==
Let $\mathbf{X}$ and $\mathbf{Y}$ be Boolean spaces. A morphism from $\mathbf{X}$ to $\mathbf{X}$ is a function $h:X\rightarrow Y$ that is continious: 
$\forall V\in\Omega_{\mathbf{Y}}\ h^{-1}[V]\in\Omega_{\mathbf{X}}$.

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]                        |second-order  |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

====Subclasses====
  [[...]] subvariety

  [[...]] expansion


====Superclasses====
  [[...]] supervariety

  [[...]] subreduct


====References====

[(Ln19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

Trace: • boolean_algebras • finite_commutative_binars • cancellative_commutative_monoids • stably_compact_spaces • m-zeroid • abelian_lattice-ordered_groups • abelian_groups • right_hoops • m-sets • commutative_monoids