Studia Scientiarium Mathematicarum Hungarica 26. (1991)
1. szám - Deák J.: On bitopological spaces II
■i- Studia Scientiarum Mathematicarum Hungarica 26 (1991), 1—17 ON BITOPOLOGICAL SPACES II J. DEÁK In this paper, we investigate the relations between multifunctions and bitopological separation properties. § 5 contains the definitions and gives, in terms of (families of) multifunctions into topological spaces, conditions guaranteeing that a bispace is S;, respectively that one of its topologies is Sf with respect to the other (i = 1,2, 3). (The results for i — 3 are cited from Smithson [12].) Only a few of these conditions can be shown to be necessary and sufficient, so several problems remain open. § 6 gives a complete answer ro the same question with i—n. § 7 contains some results on multifunctions between bispaces. § 8 deals with a special case of multifunctions into topological spaces, namely the decompositions of spaces. For §§ 0...4, see the first part of this series [5]; notions defined there will be used without explanation. Acknowledgement. The author wishes to thank Prof. M. Bognár (Budapest) for raising the problem of inducing bitopologies by decompositions of topological spaces. § 5. Bitopologies induced by families of multifunctions into topological spaces 5.0 Let X and Y be sets. A function in assigning to each X a subset of Y is a multifunction (multivalued function, set-valued function) from X into Y. If we are given a topology J on f [and also a topology Sfi on X], we shall refer to in as a multifunction[from the space (X, SYj] into the space (F, .T). A multifunction m into a topological space will be called compact valued, respectively closed valued if for each x£X, m(x) is compact, respectively closed. For AczY and BczX, put = {xCA": m(x)(T/4 ^ 0}, m(B) — Urn [21]. m is onto if m(Z)=F. Remarks, a) The formula xrmy *>F€m(x) establishes a one-to-one correspondence between multifunctions from X into F and relations between elements of X and F, so we could just as well speak of relations instead of multifunctions. It is, however, more in keeping with the traditions to 1980 Mathematics Subject Classification. Primary 54E55; Secondary 54C60, 54D10, 54D15, 54E99. Key words and phrases. Bitopology, (lower/upper) semicontinuous multifunction, compact/ closed valued multifunction, separation axioms, completely regular, internal proof, (orderly) (pseudo-) direction, normal pseudo-direction, compatible directional structure, separation of sets, decomposition of a space. 1 Akadémiai Kiadó, Budapest MAGYAR TUDOMÁNYOS AKADÉMIA KÖNYVTÁRA