Quotients of Proximity Spaces

A characterization of the quotient proximity is given. It is used to find necessary and sufficient conditions for every proximity map on a space to be a topological quotient map. It is shown that a separated proximity space is compact iff every /7-map on X with separated range is a proximity quotient map. Introduction. In 1959 Katetov [3] introduced proximity quotient maps. They have since been studied by Nachman [6], Poljakov ([7] and [8]), and Stone [9]. Although there are characterizations of proximity quotient maps in the literature [6], the only explicit formulation of the quotient proximity the author knows of is due to A. H. Stone, whose work appears in [10]. We give another characterization and use it to study mapping properties of proximity spaces. Our notation will follow [10]. In particular, A<^<=-B will mean A $ (X—B) and proximity maps will be called /»-maps. (X, à) will always denote a (not necessarily separated) proximity space. Given a completely regular space X, <50 will represent the fine proximity : A $0 B iff there is some/e C*(X) such that/(^)=0 and/(5)=l. 1. Characterization. 1.1 Definition. If y and <5 are two proximities on a set X, y is said to be finer than ô if A y B implies A ô B. This will be written <5<y. 1.2 Definition. Let /be a function from a proximity space (X, Ô) onto a set Y. The quotient proximity is the finest proximity on Y such that/is a/»-map. When Y has the quotient proximity, /will be called a /»-quotient map. 1.3 Theorem (Stone [9]). The quotient proximity is given by: Ce c D iff for each binary rational s e [0, 1], there is some CSS Y such that C0=C, Cx=D ands<t impliesf~x(Cs)c cf-i(Ct). Presented to the Society, December 27, 1971 ; received by the editors March 8,1972. AMS (MOS) subject classifications (1970). Primary 54E05, 54E10; Secondary 54B15.