Weak convergence of finite Borel measures in a completely regular topological space
X is defined by means of the class of bounded and continuous functions
f
:
X
→
R
. We give conditions equivalent to ...weak convergence of finite measures in terms of some classes of unbounded, continuous and semicontinuous real-valued functions on
X. For this purpose we introduce the notion of almost uniformly integrable mappings with respect to a directed family of measures. The obtained results may be treated as an extension of the Alexandroff theorem, also known as portmanteau theorem.
The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (𝑋, 𝜏) and in the family generated by the sets having the Baire property and ...given a proper 𝜎-ideal containing 𝜏-meager sets. The regularity property of such topologies is investigated.
Bornologies axiomatize an abstract notion of bounded sets and are introduced as collections of subsets satisfying a number of consistency properties. Bornological spaces form a topological construct, ...the morphisms of which are those functions which preserve bounded sets. A typical example is a bornology generated by a metric, i.e. the collection of all bounded sets for that metric. In a recent paper E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547–1556 the authors noted that many examples are known of natural functors describing the transition from categories of metric spaces to the “metrizable” objects in some given topological construct such that, in some natural way, the metrizable objects generate the whole construct. These constructs can be axiomatically described and are called metrically generated. The construct of bornological spaces is not metrically generated, but an important large subconstruct is. We also encounter other important examples of metrically generated constructs, the constructs of Lipschitz spaces, of uniform spaces and of completely regular spaces. In this paper, the unified setting of metrically generated theories is used to study the functorial relationship between these constructs and the one of bornological spaces.
The first aim of this paper is to introduce and to study the concepts of ‘complete Scott continuity’ and ‘completely induced
L-fuzzy topological space’. The second is to discuss the connections ...between some separation, countability and covering properties of an ordinary topological space and its corresponding completely induced
L-fuzzy topological space.
We show in this paper that a topological space satisfies $T_3$ (which we do not intend to imply $T_2$) if and only if convergence of filters is a continuous relation. In particular, a Hausdorff space ...is regular if and only if convergence of filters is a continuous mapping. We propose a new, categorically motivated, definition of continuous relations between topological spaces, and we compare it with two existing continuity concepts for relations.
The existence of coincidence and fixed points for continuous mappings on pseudo-compact completely regular topological spaces are proved. Our results are different from known, or are generalizations, ...extensions and improvements of the corresponding results due to Jungck, Liu and Liu et al. Further, the Edelstein result for contractive mappings is extended to Hausdorff (not necessarily completely regular) topological spaces and generalized in many aspects. An example is presented to show that our results are genuine generalizations of the Edelstein result.
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