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1.
《Quaestiones Mathematicae》2013,36(3-4):505-515
Abstract The notion of diminishing orbital diametral sum for a pair of commuting mappings has been introduced into a Hausdorff locally convex space whose topology is generated by a family of seminorms. Subsequently, this notion is utilized to prove common fixed point theorems which generalize certain theorems of Kirk and Tan. 相似文献
2.
We provide some results, in a unified way, concerning the Hausdorff, Attoch-Wets, Vietoris, Fell, Wijsman topologies, defined on the closed subsets of a metric space and the Mosco topology defined on the closed convex subsets of a metric space. In particular we analyze countability axioms, metrizability and complete metrizability. Also, we give necessary and sufficient conditions for their pairwise coincidence.Support by MURST and by the mathematics departments of the universities Limoges and Perpignan (R. L.) are gratefully acknowledged. The authors are also grateful to the referees and to G. Beer for valuable comments. 相似文献
3.
Antonio Martinón 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):3108-3111
We define the Hausdorff measure of nonconvexity β(C) of a nonempty bounded subset C of a Banach space X as the Hausdorff distance of C to the family of all the nonempty convex bounded subsets of X. We compare the measure β with the Eisenfeld-Lakshmikantham measure of nonconvexity α and prove that the two measures are equivalent (β≤α≤2β), but in general they are different. 相似文献
4.
Positivity - For a compact convex subset K of a locally convex Hausdorff space, a measurement on A(K) is a finite family of positive elements in A(K) normalized to the unit constant $$1_K , $$... 相似文献
5.
Given two compatible metrics on a metrizable space X. It is well known that they give rise to the same Hausdorff hypertopologies and upper Hausdorff hypertopologies, on the collection of all closed subsets of X, if and only if they are uniformly equivalent. This is no longer true for the lower Hausdorff hypertopology; indeed a weaker condition is needed, and this condition has been found by Costantini and Vitolo. 相似文献
6.
We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following
four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected,
Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to
give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate
the necessity of various assumptions in our results. 相似文献
7.
In this work we expand upon the theory of open ultrafilters in the setting of regular spaces. In [E. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981) 1-45], van Douwen showed that if X is a non-feebly compact Tychonoff space with a countable π-base, then βX has a remote point. We develop a related result for the class of regular spaces which shows that in a non-feebly compact regular space X with a countable π-base, there exists a free open ultrafilter on X that is also a regular filter.Of central importance is a result of Mooney [D.D. Mooney, H-bounded sets, Topology Proc. 18 (1993) 195-207] that characterizes open ultrafilters as open filters that are saturated and disjoint-prime. Smirnov [J.M. Smirnov, Some relations on the theory of dimensions, Mat. Sb. 29 (1951) 157-172] showed that maximal completely regular filters are disjoint prime, from which it was concluded that βX is a perfect extension for a Tychonoff space X. We extend this result, and other results of Skljarenko [E.G. Skljarenko, Some questions in the theory of bicompactifications, Amer. Math. Soc. Transl. Ser. 2 58 (1966) 216-266], by showing that a maximal regular filter on any Hausdorff space is disjoint prime.Open ultrafilters are integral to the study of maximal points and lower topologies in the partial order of Hausdorff topologies on a fixed set. We show that a maximal point in a Hausdorff space cannot have a neighborhood base of feebly compact neighborhoods. One corollary is that no locally countably compact Hausdorff topology is a lower topology, which was shown previously under the additional assumption of countable tightness by Alas and Wilson [O. Alas, R. Wilson, Which topologies can have immediate successors in the lattice of T1-topologies? Appl. Gen. Topol. 5 (2004) 231-242]. Another is that a maximal point in a feebly compact space is not a regular point. This generalizes results of both Carlson [N. Carlson, Lower upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007) 619-624] and Costantini [C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008) 187-225]. 相似文献
8.
The dimension print is a concept which contains more detailed information than the usual Hausdorff dimension. So, for example, a sphere and the surface of a cube have same dimension but different dimension prints. Can anything be said about the dimension print of most convex surfaces (in the Baire category sense)? 相似文献
9.
In this paper we use the upper semifinite topology in hyperspaces to get results in normal Hausdorff topology. The advantage of this point of view is that the upper semifinite topology, although highly non-Hausdorff, is very easy to handle. By this way we treat different topics and relate topological properties on spaces with some topological properties in hyperspaces. This hyperspace is, of course, determined by the base space. We prove here some reciprocals which are not true for the usual Vietoris topology. We also point out that this framework is a very adequate one to construct the ?ech-Stone compactification of a normal space. We also describe compactness in terms of the second countability axiom and of the fixed point property. As a summary we relate non-Hausdorff topology with some facts in the core of normal Hausdorff topology. In some sense, we reinforce the unity of the subject. 相似文献
10.
A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces. 相似文献
11.
Nathan Carlson 《Topology and its Applications》2007,154(3):619-624
In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ?σ in which there is no Hausdorff topology μ satisfying σ?μ?τ. τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies. 相似文献
12.
Horst Herrlich 《Applied Categorical Structures》1994,2(1):101-105
In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).Dedicated to my friend Nico Pumplün on his sixtieth birthday 相似文献
13.
Klaus D. Schmidt 《Acta Appl Math》1986,5(3):209-237
Rådström's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Rådström's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådström's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas. 相似文献
14.
A. Bella 《Topology and its Applications》2006,153(11):1651-1657
We construct under [CH] a Tychonoff pseudocompact Fréchet space and a countably compact Hausdorff Fréchet space which are both not strongly Fréchet. 相似文献
15.
Miroslav Hušek 《Topology and its Applications》2008,155(14):1493-1501
Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov. More details are devoted to the unpublished Hausdorff's approach that is shown to work in a more general situation, too. 相似文献
16.
We show that the thin geometric realization of a simplicial Hausdorff space is Hausdorff. This proves a long-standing conjecture of Graeme Segal stating that the thin geometric realization of a simplicial k-space is a k-space. 相似文献
17.
S. Dancs P. Medvegyev Gy. Magyarkuti 《Journal of Optimization Theory and Applications》2011,150(3):675-682
The purpose of this short technical note is to show that a locally convex topological vector space is normable, if and only
if an important convergence theorem about closed and convex sets holds. This new assumption of normability is related to the
problem of preservation of Hausdorff lower continuity of the intersection of Hausdorff lower continuous, closed and convex
valued correspondences. 相似文献
18.
R. Lowen 《Topology and its Applications》1981,12(1):65-74
Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces. 相似文献
19.
A.P. Kombarov 《Topology and its Applications》2010,157(4):694-697
Let X be a Hausdorff topological space and exp(X) be the space of all (nonempty) closed subsets of a space X with the Vietoris topology. We consider hereditary normality-type properties of exp(X). In particular, we prove that if exp(X) is hereditarily D-normal, then X is a metrizable compact space. 相似文献
20.
We prove that a Hausdorff space X is very I-favorable if and only if X is the almost limit space of a σ-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very I-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces. 相似文献