The first aim of this paper is to introduce and to study the concepts of complete scott continuity and completely induced lfuzzy topological space. Bishops notion of a function space, here called a bishop space, is a constructive functiontheoretic analogue to the classical settheoretic notion of a topological space. This result characterizes a completely regular space as a topological space that admits an admissible family of open coverings. We shall extend the definition of complete regularity given in 8 for hausdorff spaces to arbitrary spaces by defining a convergence. If is a completely regular space and is a subset of, then is completely regular with the subspace topology. A note on regular and completely regular topological spaces michael j. Computably regular topological spaces klaus weihrauch university of hagen, hagen, germany. It is locally compact, it is a cspace, it is a topological lattice. If you have a particular space in mind, like say, the real line with the usual topology, then you should probably rephrase your quest. Informally, 3 and 4 say, respectively, that cis closed under. However, there is another useful and natural approach to defining completeness in a topological space. I be a family of completely regular spaces then the. Since any two separated sets are semiseparated, every separated space is semi separated.
Completely induced lfuzzy topological spaces sciencedirect. If uis a neighborhood of rthen u y, so it is trivial that r i. A subset uof a metric space xis closed if the complement xnuis open. Continuous orderpreserving functions on a preordered. They form one of the most important classes of topological spaces, which is distinguished by several special properties and is very often encountered in the applications of topology to other branches of mathematics. Every space which is completely regular is also regular, since, for example, f 10. Raha 1 proceedings of the indian academy of sciences mathematical sciences volume 102, pages 49 51 1992 cite this article. Indeed, in the context of mathematical psychology, it is often assumed that a set. An example of a regular space that is not completely regular a. Why say completely regular t1 when completely regular t0.
In a regular t 3 space, 1point sets are closed and for and closed not containing, there exist disjoint open sets containing and. It is common to place additional requirements on topological manifolds. Example of a completely regular spaces mathoverflow. An example of a regular space that is not completely regular. A topological space x is a t 1 space if it satisfies the first axiom of separation and a hausdorff t 2 space if it satisfies the second axiom of separation. Then every sequence y converges to every point of y. Topological spaces dmlcz czech digital mathematics library. A topological space x is called locally euclidean if there is a nonnegative integer n such that every point in x has a neighbourhood which is homeomorphic to real nspace r n. A topological manifold is a locally euclidean hausdorff space. Topological spaces x for which cx is a dual ordered vector space. For brycs inverse varadhan lemma, however, the complete regularity of the topological space is needed see dembo and zeitouni, 1993, chap.
A topological space x is said to be completely regular if it is t 1 and for any point x 0 2x and any closed subset a x such that x. Levine 36 introduced the concept of simple expansion and in 1965 37 discussed spaces in which the compact and closed sets are the same maximal compact spaces proving that the product of a maximal compact space with itself is maximal. The theory of measures in a topological space, as developed by v. Pdf completely regular fuzzifying topological spaces. Topological spaces, bases and subspaces, special subsets, different ways of defining topologies, continuous functions, compact spaces, first axiom space, second axiom space, lindelof spaces, separable spaces, t0 spaces, t1 spaces, t2 spaces, regular spaces and t3 spaces, normal spaces. Sandwichtype characterization of completely regular spaces. A topological ordered space is a triple x,, where is a topology on x and is a partial order on x. All the higher separation axioms in topology, except for complete. In topology and related branches of mathematics, tychonoff spaces and completely regular spaces are kinds of topological spaces. Since any two separated sets are semi separated, every separated space is semi separated. Introduction in chapter i we looked at properties of sets, and in chapter ii we added some additional structure to a set a distance function to create a pseudomet. Show that a regular space need not be a hausdorff space.
A completely regular topological space x is separable and metrizable if and only if ccx is second countable. In addition, the admissible family of coverings provides an interesting methodology of studying aspects of uniformity and. X so that u contains one of x and y but not the other. Needless to say, the assumption of complete regularity is particularly interesting in decision theory since, for example, it is well known that each topological group is a completely regular space see e. A simpler example of regular space that is not completely regular is attempted. Text or symbols not renderable in plain ascii are indicated by. A t 1space is a topological space x with the following property. Ais a family of sets in cindexed by some index set a,then a o c. Characterizations of the completely regular topological spaces. Regular spaces that are not completely regular mathoverflow. A completelyregular hausdorff space is called cech complete if it can be represented as the intersection of a countable family of open sets in a certain hausdorff compactification. Completelyregular space encyclopedia of mathematics.
Let x be a completelyregular topological space and le. Introduction in 1925 urysohn 10i posed, but left unanswered, the question of whether or not regular topological spaces exist in which every continuous realvalued function is constant. This isnt, to my knowledge, an actual term meaning anything. To further justify the above topological setting, notice that in a regular topological space y, the rate function associated with the ldp is unique and varadhans integral lemma is applicable. Every t0strongly topological gyrogroup is completely regular. Varadarajan for the algebra c of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary. Co nite topology we declare that a subset u of r is open i either u. For instance, the space of any topological group is a completelyregular space, but need not be a normal space. Extensive research on generalizing closedness was done in. Katsaras received 24 march 2005 and in revised form 22 september 2005 some of the properties of the completely regular fuzzifying topological spaces are investigated. Furthermore, we note that a completely regular ordered ispace is strictly completely regular ordered provided that it satisfies at least one of the following three conditions.
A t 1 space is a topological space x with the following property. Let gbe a topological group, let 1 g denote the identity element in g. How to verify whether a usual topological space is. We present an example of a completely regular ordered space that is not strictly completely regular ordered. Y is a retract of x, and y is a deformation retract of x. It is a different example from that in steen and seebach or dugundji for that matter, in that it doesnt use ordinal numbers. Next, we show that every t0strongly topological gyrogroup g is a microassociative hausdor. To further justify the above topological setting, notice that in a regular topological space y, the rate function associated with. Let fr igbe a sequence in yand let rbe any element of y.
The condition of regularity is one of the separation axioms satsified by every metric space and in this case, by every pseudometric space. A topological space is said to be absolutely closed provided it is closed in every extending space. Some mo tivation for this terminology lies in the fact that a t1 topological space is separated iff it is hausdorff. We then looked at some of the most basic definitions and properties of pseudometric spaces. Free topology books download ebooks online textbooks. The second is to discuss the connections between some separation, countability and covering properties of an ordinary topological space and its corresponding completely induced lfuzzy topological space.
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