Studia Scientiarium Mathematicarum Hungarica 26. (1991)
1. szám - Deák J.: On bitopological spaces II
BITOPOLOGICAL SPACES II 3 d) In place of 9W, it would be more precise to take {(m,-, $~t)\ idl). We have chosen, however, the simpler terminology and assume instead that whenever e) As the results of this paper are evidently valid for the empty bispace, we may always assume in the proofs that X?± 0. 5.2 If arbitrary multifunctions into arbitrary spaces are allowed then each bitopology can be induced by a family of multifunctions. Indeed, put Y= {1, 2}, ^{{1}} andY if *€G _ íY if x£F {2} if x$G W “ |{1} if x$F 9JÍ = {mG: 0 G’€^}U{mF: 0 * F£co-J}. Then X = Xffl,. The problem is more complicated if the multifunctions tn and/or the topologies 3Tm are supposed to satisfy some conditions. The cardinality of 9JÍ can also be restricted; in particular, it is an interesting question which bitopologies can be induced by a single multifunction (satisfying certain conditions). Let us start the investigation of such problems with some remarks: a) One can usually assume without loss of generality that m(x)?±0 (x£Z) [equivalently: m-1(Fm)=Ar]. Indeed, if this condition is not satisfied then take the topological sum of Y and a one-point space {z} and put n(x)=m(x)U {z}. Now ti (x)^0 (x£X) and X„=Xm. What is more, n and inherit from tn and 2Tm most of their good properties, in fact all the properties considered in what follows (multifunctions will be closed valued or compact valued, topological spaces will be compact and/or they will satisfy one of the usual separation axioms). b) In contrast, supposing m to be onto is a real restriction. We can make tn onto by substituting m(A') for Ym, but in this case the non-hereditary properties of STm (e.g. compactness) are lost. c) Let (Z, be the T0-identification of (Fm, Tf) and n(x) = {z€Z: m(x)ilz 0} (z£Z). Then X„=Xm and n is compact valued, closed valued, respectively onto if tn has the same property. Consequently, the topologies 3Tm can always be assumed T0. d) Each family of multifunctions can be replaced by a single multifunction inducing the same bitopology. Let (T, .5") be the topological sum of the spaces (ym, 3~m) (mCäJl) and Z—Y\J{w} where w£ Y. To simplify the notations, assume that the sets Ym are disjoint and Y= U {Fm: m€9Jl}. Put "(*) = ( U m(x))U{w} (x€T) and let A£.2Tn iff A\{w}£&~ and either w $ A or A covers all but a finite number of the sets Tm. Then X„— Xtpi, furthermore, rt and STn inherit many good properties of the multifunctions rnCSÖÍ and of the topologies 2Tm. In particular, c) and d) give: Theorem. Each bitopology can be induced by a single compact valued multifunction onto a compact normal T0-space.