Studia Scientiarium Mathematicarum Hungarica 26. (1991)

1. szám - Deák J.: On bitopological spaces II

J. DEÁK work with multifunctions, although some authors prefer the relations. In addition, if the definition below were formulated for relations, it would be in conflict with the usual definition of lower/upper semicontinuity in the special case when (Y, 2~)= = (R, S) and rm happens to be a function. b) The notations and m(B') are motivated by the equalities m_1(/l}= = r^[A] and m(F) = /•„,[£]. c) Closed valued multifunctions are sometimes called point closed. Definition (Wilson [15], Kuratowski [8] and Bouligand [3]). A multifunction m from a space (X, Sf) into a space (Y, ,T) is a) lower semicontinuous if for each ^-open set G, trt-1(G) is .S^-open; b) upper semicontinuous if for each ^-closed set F, tn_1(F) is ^-closed. Remarks, d) For the motivation of the names, see e.g. [7] 1.7.17 (a). e) Kuratowski gave these definitions only for closed valued multifunctions into compact metrizable spaces. (Wilson considered an even more special case.) The restriction on Y was later dropped, but upper semicontinuous multifunctions are often closed valued or compact valued by definition; the same applies sometimes also to lower semicontinuity; see e.g. [13], [2], [6], [7], and also the references in [13]. f) It is often contained in the definition of lower/upper semicontinuity (or even in the definition of a multifunction) that m(x)^0 for each x£X, cf. [2], the foot­note on p. 114. g) Sometimes other names are used instead of lower/upper semicontinuity, e.g. infra- and supra-continuity in [15]; cf. [13]. 5.1 Definition (Smithson [12]). Let X be a set and 9JÍ a family of multifunctions m from X into topological spaces (Ym, ,Tm). Let and be the coarsest topologies on X making each m£äR lower, respectively upper semicontinuous. Then (!Pm, 2m) is the bitopology induced by SR. For a single multifunction m, (^{m}, J{m}) is called the bitopology induced by m. Notations. 9>m = 0>ln), 3m = 2(m), XOT = (T; 2W), Xnt = (X; 0>m, Jm). Remarks, a) The systems (m_1(G>: G€^-m,m€SR} and {m-1(F>: JL\F€^n,m6SR} form a subbase for respectively a closed base for . In the case of it is enough to take inverse images of open bases (but usually not subbases); in the case of £m, it is not enough to consider inverse images of closed bases. b) = sup {£?m: m£SR}, = sup {ÜOT: mgifil}. c) We have interchanged the role of á3 and 2. in the above definition in order to adjust it to our other definitions,1 e.g. R. 1 Added in proof. Cf. the footnote to 0.4.

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