Open mappings on extremally disconnected compact spaces. (English) Zbl 1230.54029

Let \(B\) be a complete Boolean algebra and let \(\kappa\) be a regular cardinal. A \(\kappa\)-complete uniform retract of \(B^\kappa\) is a \(\kappa\)-complete homomorphism \(r:B^\kappa\to B\) such that \(r(\langle b,b,\dots\rangle)=b\) for \(b\in B\) and \(r(\langle b_\alpha\rangle_{\alpha<\kappa})=0\) if \(|\{\alpha<\kappa:b_\alpha\neq0\}|<\kappa\). For \(\kappa=\omega\), \(\kappa\)-complete uniform retracts exist by Sikorski’s Extension Theorem and for \(\kappa>\omega\), \(\kappa\) must be inaccessible. Let \(r:B^{[\kappa]^{<\omega}}\to B\) be a \(\kappa\)-complete uniform retract. Then \(\text{Exp}_\kappa(B,r)=\{T\in B^{[\kappa]^{<\omega}}: (\forall s\in[\kappa]^{<\omega})\) \(r(\langle T(s\cup\{\alpha\})\rangle_{\alpha<\kappa})=T(s)\}\) is a complete algebra which is a \(\kappa\)-complete subalgebra of \(B^{[\kappa]^{<\omega}}\).
These complete Boolean algebras generalize the complete Boolean algebras known from Mathias forcing, Prikry forcing and forcing notions with the same combinatorial property known as the Prikry property. The canonical shift operation on \(\text{Exp}_\kappa(B,r)\) is defined by \(\text{sh}(T)(\emptyset)=T(\emptyset)\) and \(\text{sh}(T)(s)=T(s\setminus\{\min s\})\) if \(s\neq\emptyset\) is a complete homomorphism such that (i) if \(T\neq0\), then there is a nonzero \(S\leq T\) such that \(S\cap\text{sh}(S)=0\), (ii) there is no partition \((T_0,T_1,T_2)\) of unity such that \(T_i\cap\text{sh}(T_i)=0\) for all \(i=0,1,2\).
Via the Stone duality the author obtains an extremally disconnected compact space and an open mapping on it such that the set of fixed points is nowhere dense but not empty. The main theorem of the paper states the opposite: Any such topological situation is essentially the result of such a construction.


54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
03E40 Other aspects of forcing and Boolean-valued models
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