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Equationally closed subframes and representation of quotient spaces. (English) Zbl 0789.54008
This article studies the equationally closed subframes of a frame and their relationship to quotient spaces in the category of topological spaces. If \(A\) is a frame, a subframe \(B\) is called an equationally closed subframe (EQC) iff for all \(a\in B\), \(x \in A\), if \(a \wedge x \in B\) and \(a \vee x \in B\), then \(x \in B\). More generally, a frame homomorphism \(\theta:C \to A\) is equationally closed iff \(\theta(C)\) is EQC in \(A\). Some properties of these notions are developed. Among other things it is shown that subframes which are sub-Heyting algebras are automatically EQC, both open and closed frame homomorphisms are EQC and a subframe \(C \subseteq A \times A\) is EQC iff \(C\) is a congruence.
Recall that to every topological space, we can associate the frame \(\Omega(X)\) of its open subsets. If \(f:X \to Y\) is a continuous map of spaces, then the inverse image map \(f^{-1}=\Omega(f):\Omega(Y) \to \Omega(X)\) is a frame homomorphism. Also, recall that a space \(X\) is said to satisfy the separation axiom \(T_ D\) iff each \(x\in X\) has an open neighborhood \(U\) such that \(U-\{x\}\) is open. These are the spaces \(X\) such that applying the functor \(\Omega\) to onto maps with codomain \(X\) results in a monomorphism of frames. The main result is that a space \(Y\) satisfies \(T_ D\) together with a condition called approximation by closed sets iff the quotient mappings \(f:X \to Y\) are precisely those continuous maps with \(\Omega(f)\) an EQC monomorphism. The class of such spaces includes the Fréchet (and hence metrizable) spaces.

MSC:
54B15 Quotient spaces, decompositions in general topology
54B30 Categorical methods in general topology
06D20 Heyting algebras (lattice-theoretic aspects)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
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