## 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)
Full Text:

### References:

 [1] C.E. Aull and W.J. Thron : Separation axioms between T0 and T1 , Indag.Math. 24 ( 1962 ), 26 - 37 MR 138082 | Zbl 0108.35402 · Zbl 0108.35402 [2] B. Banaschewski and A. Pultr : Variants of openness, Seminarberichte aus dem Fachbereich Mathematik , FernUniversität Hagen , Nr. 44 - 1992 , Teil 1 , 39 - 54 MR 1300720 [3] E. Čech: Topological Spaces, Publishing House of the Czechoslovak Academy of Sciences , Prague , 1965 MR 211373 | Zbl 0141.39401 · Zbl 0141.39401 [4] D. Dikranjan and E. Giuli : Closure operators induced by topological epireflections , Coll.Math.Soc.János Bolyai 41 ( 1983 ), 235 - 24 MR 863906 | Zbl 0601.54016 · Zbl 0601.54016 [5] D. Dikranjan , E. Giuli and W. Tholen : Closure Operators II, Categorical Topology (Proceedings of the Conference , Prague 1988 ), World Scientific Publ . Co. 1989 , 297 - 335 MR 1047909 [6] H. Herrlich and G.E. Strecker : Category Theory , Allyn and Bacon , 1973 MR 349791 | Zbl 0265.18001 · Zbl 0265.18001 [7] P.T. Johnstonc : ” Stone Spaces ”, Cambridge University Press , Cambridge , 1982 MR 698074 | Zbl 0499.54001 · Zbl 0499.54001 [8] A. Joyal and M. Tierney : An extension of the Galois theory of Grothendieck , Memoirs of the AMS , Volume 51 , Number 309 (September 1984 ) MR 756176 | Zbl 0541.18002 · Zbl 0541.18002 [9] S. MacLane : Categories for the Working Mathematician , Springer-Verlag , New York ( 1970 ) MR 1712872 | Zbl 0705.18001 · Zbl 0705.18001 [10] A. Pultr and A. Tozzi : Notes on Kuratowski-Mrówka theorems in point-free context , Cahiers de Top.et Géom.Diff.Cat. XXXIII - 1 ( 1992 ), 3 - 14 Numdam | MR 1163423 | Zbl 0772.54016 · Zbl 0772.54016 [11] A. Pultr and A. Tozzi : The role of separation axioms in algebraic representation of some topological facts, Seminarberichte aus dem Fachbereich Mathematik , FernUniversität Hagen , Nr. 44 - 1992 , Teil 2, 322 - 332 [12] W J.Thron : Lattice-equivalence of topological spaces , Duke Math.J. 29 ( 1962 ), 671 - 679 Article | MR 146787 | Zbl 0109.15203 · Zbl 0109.15203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.