Mathematics

We prove that the road space of an R-special tree is contractible and that a locally metrizable space containing a copy of an uncountable ω 1 -compact subspace of a tree is not. We also raise some questions about possible generalizations.

Here we have introduced the ideas of (j?�i)s g ??κ -closed sets and a semi generalized closed set in a bispace; i,j=1,2;i?�j and then have studied on pairwise semi T 0 -axiom, pairwise semi T 1 -axiom and pairwise semi T ? -axiom. We have investigated some of their topological properties and also established a relation among these axioms under some additional conditions.

A topological space X is strongly D if for any neighbourhood assignment { U x :x∈X} , there is a D⊆X such that { U x :x∈D} covers X and D is locally finite in the topology generated by { U x :x∈X} . We prove that ♢ implies that there is an HF C w space in 2 ω 1 (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly D . We also show that any HFC space X is dually discrete and if additionally, countable sets have Menger closure then X is a D -space.

We show that a surjective map between compact ANR's (absolute neighborhood retracts) is a homotopy equivalence if the fibers are contractible and either the domain is simply connected or the fibers are also ANR's. This is a geometric analogue of the Vietoris-Begle theorem. We use it to show that if L is a locally convex Riesz space, C⊂L is compact, convex, and metrizable, x∈L , and the function y↦x∨y ( y↦x∧y ) is continuous, then the {x∨y:y∈C} is a compact contractible ANR.

Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.

We pose the following new variant of the Kuratowski closure-complement problem: How many distinct sets may be obtained by starting with a set A of a Polish space X , and applying only closure, complementation, and the d operator, as often as desired, in any order? The set operator d was studied by Kuratowski in his foundational text \textit{Topology: Volume I}; it assigns to A the collection dA of all points of second category for A . We show that in ZFC set theory, the answer to this variant problem is 22 . In a distinct system equiconsistent with ZFC, namely ZF+DC+PB, the answer is only 18 .

In this article we compute the number of fuzzy bitopological space with having two open sets, three open sets, four open sets and five open sets. Also, we have given some results on computation of number of fuzzy bitopological space.

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

We study point-separating function sets that are minimal with respect to the property of being separating. We first show that for a compact space X having a minimal separating function set in C p (X) is equivalent to having a minimal separating collection of functionally open sets in X . We also identify a nice visual property of X 2 that may be responsible for the existence of a minimal separating function family for X in C p (X) . We then discuss various questions and directions around the topic.

The goal of this report is to investigate the variety of Hausdorff compactifications of R . The Alexandroff one-point compactification, the two-point compactification, and the Stone-Cech compactification are all clearly different. The ultimate aim is to show that there are in fact uncountably many. An intermediate aim is to exhibit one compactification of R different from all the compactifications already mentioned.