This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental ...

Unilateral closure of maxillary extraction spaces in patients with Class III malocclusion can be challenging. This case report describes the closure of first premolar and first molar extraction spaces in a patient with a Class III dental relationship. Two miniscrews were used for intraoral skeletal anchorage. The Class III dental relationship was corrected; a positive vertical overbite was achi...

Here we consider finite-dimensional vector spaces and linear operators on them. Some of the basic ingredients will be motivated by ∗-algebras of linear transformations on inner product spaces, as in [Arv, Dou]. In particular, let us note that in the theory of C and Von Neumann algebras, the “double commutant” of a ∗-algebra is a kind of basic closure of the algebra. A special case concerns alge...

In this article, we formalize topological properties of real normed spaces. In the first few parts, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. In the middle of the article, we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, ima...

The present paper introduces the notion of the complete fuzzy norm on a linear space. And, some relations between the fuzzy completeness and ordinary completeness on a linear space is considered, moreover a new form of fuzzy compact spaces, namely b-compact spaces and b-closed spaces are introduced. Some characterizations of their properties are obtained.

The purpose of this paper is to introduce and study regular bopen sets (briefly rb-open sets) in topological spaces and obtain some of their properties. We introduce the rb-closure and discuss some basic properties of the rb-closure. Also we study the new concepts of rb-closed spaces by means of filter bases. Mathematics Subject Classification: 54A05, 54A20

In this paper, we introduce new definitions of L-fuzzy neighborhood systems, L-fuzzy interior operators and L-fuzzy closure operators. Three characterizations of the category L-FTOP of L-fuzzy topological spaces and their L-fuzzy continuous mappings are presented by means of the category L-FNS of L-fuzzy neighborhood spaces and their continuous mappings, the category L-FIS of L-fuzzy interior s...

This technical report summarized facts from the basic theory of generalized closure spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Set-Valued Set-Functions 1.1. Closure, Interior, Neighborhood, and Convergent. In this section, which in part generalizes the results o...

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicorefle...