The Wallman compactification is an epireflection

Abstract

It is shown that a map having an extension to a closed map between the Wallman compactifications of its domain and range has a unique such extension. A consequence is that the collection of such maps forms the morphisms of a category on which the Wallman compactification is an epireftection, answering a question raised by Herrlich. We establish the existence of a category C of spaces and maps on which the Wallman compactification is an epireflection functor. This answers a question raised by Herrlich in [1] as to whether such a category exists. A space is a T1 topological space and a map is a continuous function. For categorical terminology see [1]. The class of objects of C will be the class of spaces, and the morphisms will be the class of maps f:X-Y such that there is a closed map g: wX --w Y with gwx = wyf, where wx, wy are the inclusions of X, Y into their Wallman compactifications wX, w Y. Any such closed map g is called a w-extension off It is clear that for each Xe C the maps lx and wx are morphisms, since 1wx is a w-extension of these maps. Also trivially if f is a morphism and g is a w-extension of f then g is a morphism (being its own w-extension). Finally since the composition of closed maps is a closed map it follows that the composition of morphisms is a morphism. To show that the Wallman compactification is an epireflection on this category we need only show that each morphism has a unique w-extension. To do this we examine the construction of the Wallman compactification of X as given for example in [2]. The points of the Wallman compactification are the maximal closed filters [p] on X; we write [p] for the filter and p for the point of wX. For each p E wX there is the open filter (p) consisting of the open subsets of Y which contain a member of [p]. The neighborhood filter of p E wX is Received by the editors February 10, 1971. AMS 1970 subject class/ilcations. Primary 54D35.