Arhangel’skĭı [3] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper. The concept of countable core in [3] is a little hard to understand at first; Arhangel’skĭı, however, provides equivalents which are easier to understand, and so we will take one of...

This paper is the second part of the paper [2]. Both of them are in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations ...

1 Topological Spaces 1 1.1 Continuity and Topological Spaces . . . . . . . . . . . . . . . 1 1.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Further Examples of Topological Spaces . . . . . . . . . . . . 3 1.5 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Hausdorff ...

This paper investigates the fractals generated by the dynamical system of intuitionistic fuzzy contractions in the intuitionistic fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard intuitionistic fuzzy metric spaces by using the intuitionistic fuzzy Banach contraction theorem. In addition to that, we an...

Definition 1 (compare [2,5,8]) A topological space is called a KC-space provided that each compact set is closed. A topological space is called a U S-space provided that each convergent sequence has a unique limit. Remark 1 Each Hausdorff space (= T 2-space) is a KC-space, each KC-space is a U S-space and each U S-space is a T 1-space (that is, singletons are closed); and no converse implicatio...

The Hausdorff–Alexandroff Theorem states that any compact metric space is the continuous image of Cantor’s ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor’s ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact counta...

Abstract We propose a method for constructing generalized fuzzy Hausdorff distance on the set of nonempty compact subsets given metric space in sence George–Veeramni and Tian–Ha–Tian. Next, we define fractal spaces. Morever, obtain fixed point theorem class contractions present an application integranl equation.

in this paper we introduce entire sequence spaces defined by a sequence of modulus functions . we study some topological properties of these spaces and prove some inclusion relations.

These are some brief notes on measure theory, concentrating on Lebesgue measure on Rn. Some missing topics I would have liked to have included had time permitted are: the change of variable formula for the Lebesgue integral on Rn; absolutely continuous functions and functions of bounded variation of a single variable and their connection with Lebesgue-Stieltjes measures on R; Radon measures on ...

Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ω ([2] and [8]). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Haus...