Ivan L. Reilly

Separation axioms in fuzzy topological spaces

Abstract This paper provides a hierarchy of separation properties for fuzzy topological spaces which are generalizations of the standard topological notions.

On semi-regularization topologies

This paper discusses several properties of topological spaces and how they are refelected by corresponding properties of the associated semi-regularization topologies. For example a space is almost locally connected if and only if its semi-regularization is locally connected. Various separation, connectedness, covering, and mapping properties are considered.

A generalized contraction principle

the limit of the sequence of Picard iterates [j (x)} for each x in X . Several authors have considered contractive mappings in the more general setting of uniform spaces, for example Brown and Comfort [2], Edelstein [4], Kammerer and Kasriel [5], Knill [7] and Naimpally [S], and various extensions of Banachs Theorem have been obtained. The purpose of this note is to present an extension of the theorem to arbitrary Hausdorff spaces. The machinery required is the concept of a quasi-gauge structure for topological space. By a quasi-pseudo-metric on a set X we mean a non-negative real valued function on X x X which vanishes on the diagonal and satisfies the triangle inequality.

Effective tutorial programmes in tertiary mathematics

Papers presented at earlier Delta conferences reported on the use of collaborative tutorials and peer-tutors in undergraduate mathematics courses at The University of Auckland. This paper reports on significant developments within the Departments programmes, including the extension of the collaborative small-class tutorials to cover all first-year undergraduate courses in the department. It provides evidence of the effectiveness of the tutorial programme, as gauged from a survey of students’ perceptions. Many of the students enrolled in the Tutoring in Mathematics course that initially trains and provides the peer-group tutors continue on as tutors in other mathematics courses within the Department. Some subsequently enrol in the Universitys Graduate Diploma in Teaching (Secondary). The effects of the tutoring course on tutors continuing to tutor at higher levels, and their performances as novice teachers in the Diploma course are examined.

A Lattice-valued Banach-Stone Theorem

Let X and Y be compact Hausdorff spaces, and let E be a Banach lattice. In this short note, we show that if there exists a Riesz isomorphismΦ:C(X, E) →C(Y, R) such that Φ(f) has no zeros if f has none, then X is homeomorphic to Y and E is Riesz isomorphic to R.

Quasi-uniform hyperspaces of compact subsets

Abstract Let (X, u ) be a quasi-uniform space, K (X) be the family of all nonempty compact subsets of (X, u ) . In this paper, the notion of compact symmetry for (X, u ) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K(X) are examined. Furthermore we establish that for a compactly symmetric quasi-uniform space (X, u ) the Bourbaki quasi-uniformity u ∗ on K (X) is complete if and only if u is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasi-uniform setting.

Characterizations of Quasi-Metrizable Bitopological Spaces

In this paper we prove that a pairwise Hausdorff bitopological space (X, 3~x, J~2} is quasi-metrizable if and only if for each point x e X and for i, j = 1,2, i # j , one can assign S] nbd bases {S(n,i; x)\n = 1,2,... } such that (i) y <£ S(n l , i ; x) implies S(n,i; x) n S(n,j\ y) = , (ii) y e S(n,i; x) implies S(n,i; y) c S(n 1, /; x). We derive two further results from this. 1980 Mathematics subject classification (Amer. Math. Soc): 54 E 55.

On essentially pairwise Hausdorff spaces

SummaryThis paper introduces a new form of the pairwiseR1 property for bitopological spaces, and investigates spaces having this property. It is shown that such spaces are essentially pairwise Hausdorff in their behaviour except in the presence of compactness.

Some properties of quasiuniform multifunction spaces

The aim of this paper is to explore some properties of quasiuniform multifunction spaces. Various kinds of completeness of the quasiuniform multifunction space ( Y mX , U mX ) are characterized in terms of suitable properties of the range space ( Y, U ). We also discuss the local compactness of quasiuniform multifunction spaces. By using the notion of small-set symmetry, the classic result of Hunsaker and Naimpally is extended to the quasiuniform setting.

Topological properties defined by games and their applications

Abstract In this paper, we study some two person games and some topological properties defined by them. As an application of our results we refine the Choquet–Dolecki theorem on multifunctions and the classical Vainstein lemma on closed maps.