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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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