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ΔΗΜΟΣΙΕΥΜΕΝΗ ΕΡΓΑΣΙΑ

BULLETIN OF THE GREEK

MATHEMATIKAL SOCIETY

Volume 53, 2007 (119-129)

QUOTIENT LOCAL UNIFORMITIES

INTRODUCTION

Quotient structures in the theory of uniform spaces have been considered by R.W. Bagley [1] and C.J. Himmelberg [5] (see also [14]). Moreover, W. Carlson [2] studied quotient structures in quasi-uniform spaces. The more general case of semi-uniform spaces [8] in connection with quotient structures was already considered in [9], [10] (cf. also [12]).

Our aim in this paper is to study quotient structures in the case of locally uniform spaces; these spaces were considered by J. Williams [15] and also, independently, by this author in [8]. Furthermore, the same notion was employed by F. Jeschek [4] and also in [12], in connection with the classical Ascoli-Arzela Theorem.

More precisely, the content of the paper is as follows: If  is a locally uniform space and R an equivalence relation on X, let  be the (canonical) quotient map. So we give in Section 1 necessary and sufficient conditions such that the "direct image" of Q,  be a local uniformity, so that it is then a quotient local uniformity, as well (see Theorem 1.1 and Remark 1.1, iii)). This (quotient) local uniformity is also the image via p of the largest saturated local uniformity contained in Q (same theorem, in connection with Proposition 1.1). In section 2, taking a locally uniform space  with X equipped with the induced topology from Q, we give sufficient conditions such that the quotient local uniformity defines the quotient topology on  (see, for instance, Theorem 2.3).

We continue in Section 3 by considering the equivalence relation  in X determined by the intersection of the members of Q the so-called "atom" of Q, a notion applied by H. Nakano [6] in the classical case of uniform spaces (see Theorem 3.1). The saturation of Q, with respect to  yields a (saturated) local uniformity in X, topologically equivalent to Q (Theorem 3.2). The respective quotient local uniformity on  endows the latter space on it the same topology as that one defined on it by the topology of Q on X (Corollary 3.1). Finally, in Section 4 we consider a locally uniform space  and the equivalence relation defined in the underlying topological space X by a group of homeomorphisms of X. So we give by Theorem 4.1 a characterization of the case that the image of Q in the corresponding quotient (orbit) space of X be the quotient local uniformity.

I wish to express my heart-left thanks to Professor A. Mallios for many stimulating and enlightening discussions, we have had during the writing of this paper. My thanks are also due to the referee for careful reading of the paper and his remarks.