Abbreviation: BSp

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 ($\forall x\ne y\in X\exists U\in\Omega (x\in X\text{ and }y\in X\setminus U\in\Omega)$).

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: $\forall V\in\Omega_{\mathbf{Y}}\ h^{-1}[V]\in\Omega_{\mathbf{X}}$.

Example 1:

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