A Note on Wallman Extendible Functions

Abstract

Abstract.It is known that any continuous function into a T¡ space has a unique continuous Wallman extension, and that any continuous Wallman extension of a continuous function with a 7*3 range must be unique. We show that for any 7"3 space Y which is not Tt there exists a Tz space .Vand a continuous function/: X-+Y which has no continuous Wallman extension.In this paper we will consider only Tx spaces. In [2] it is shown that if Y is a T3 space and/:A->-F is a continuous function having a continuous Wallman extension /*: W(X)->-WiY) then the extension is unique. Furthermore it follows immediately from the fact that if Y is F4 then IF(F) is F2 and from the Taimanov theorem (see [1, p. 110]) that if Y is F4 then any continuous function /: A->-Y has a continuous Wallman extension, and so it is natural to ask whether the condition that Y be F4 can be relaxed. In this paper we show that, if consideration is restricted to F3 spaces, the answer is no.Recall that for a given space X the Wallman compactification WiX) is the collection of all ultrafilters in the lattice of closed subsets of X given the topology generated by the collection of all sets of the form CiA) = {u £ WiX): A £ u}, where A is closed in A' as a base for the closed sets.