In this paper we study a general notion of a uniform convergence 8tructure. Morphisms of varioua supremam-complete subsets of the complete lattice of all uniform convergence structures are investigated. They provide a satisfactmy framework for the unifonnization of arbitrary convergences. C.COOK and H.FEKIEER, modifying the axioms of W m [is], introduced in 141 it notion of a uniform convergence structure. A uniform convergence structure in their sense is an ,+ideal ~t of filters on x x x containing the discrete filter of the diagonal #,(A) = {M c X x X: A c B') and closed with respect to the inverse operation(5 E u + 3-1 E u) and the composition (El, Sz E tt =$ 31 o 8% E u). If a, COOK-FISCHER uniform convergence structure is a principal +ideal then it has an &ssociated uniform structure in the Bowsense und the resulting convergence is a, completely regular topology. The theory of COOK-FISCHEB d o r m convergence struotnres does not however contain quasi-uniform structures introduced by A.Cs.is& (51 and studied by W.Pmvm [GI, M.Mmt~mmvu and 5 . N -u~ [13] and others. Quasi-uniformities satisfy all the axioms of Bourbaki but symmetry, and therefore lead to arbitrary topologies. Both C O O K -F I S~ and gnsSi-uniform structures have been subsequently generalized in many directions (see eg. (11, [9], [lo], 1121, (161 and rig]), but in general it has beenrequired, with few exceptions, that a, uniform convergence s t r u e is an A-ideal of filters containing the discrete filter of the diqonal.In this paper we study arbitrary familiea of filters on X x X and call them unifwm wnvergence &-uciureS (or UC-struccures). Thi5 setting enables one to consider the problem of uniformization for all types of convergenm which are currently being investigated (not necessarily pseudotopological or HAUSDORFIF). Moreover, this treatment of UC-structures without prejudice turned out to be gratifying, bemuse it enabled u9 to use a unified frsmework for our studies.
In this paper we examine various typea filters that appear in the context of continuow relations, pseudo-uniform spaces and pseudotopological spaces. There arise compactoid, precompct, totally bounded, bounded and other filters. We study their properties and relationship among them. Terminology and NotationsWe recall here some basic facts from convergence theory. For a systematic account of convergence and uniform convergences spaces the reader is referred to [6, 71, [3] and [l].Let X be a nonempty set. Denote by 4X the collection of all filters on X. We say that two filters 3 , b E 4X meet if for every F E 3 and Q E b, F n Q =I. 0. The supremum 3 v 3 exists if and only if 3 and b meet. Then we write 3 v b $; 2=. For a subset A of X let us denote by N , ( A ) the discrete (principal) filter of A : N , ( A ) = ( B c X : A c B}. Clearly, N , ( A ) E 4X if and only if A $; 0. Instead of JV,((x}) we will write N , ( z ) . We say that 3 meets A if 3 and N , ( A ) meet.If 3 E 4X then /W will denote the collection of all ultrafilters finer than 3. The grill 3* of 3 is the family of all subsets of X which intersects every element of 3.Assume now that X is a real linear space. Let 3, b be filters on X and 3 a filter on R. Define the filters 3 + b, 33 and conv 3 taking as their bases the families {F + Q: F E 3, CT E a}, { S F : S E 3, F E 5) and (conv F : F E 3}, respectively. Instead of N,(S) 3 we will write X3. N,(O) will denote the neighborhood filter of 0 on R with respect to the standard topology of R.Let X be a set. Every mapping n : 4X --f is called a convergence on X . The value of a convergence n at a filter Swill be denoted by Limn 3 or Lim 5. We say that a filter 3 is convergent [convergent to 2 3 if Lim 3 $; 0 [a: E Lim 3 1 . A convergence n is said to be a pseudotopology if it satisfies the following conditions : (P 1) (P 2) (P 3)If n is a pseudotopology on X then the pair (X, n) is called a pseudotoplogical space.x E Limn N , ( x ) for every x E X . 3-c $ implies Limn 3 c Limn 3. Limn 3 n Limn 3' c Limn (3 n 3').
Abstract. We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex L p (T, E
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