A note on the location of joint discontinuity (Q1066527)

In this paper S is taken (for simplicity) to be a compact, connected, semitopological monoid whose minimal ideal is a group. Let X be a compact space, and consider a separately continuous action \(\mu\) : \(X\times S\to X\) of S on X. This article is concerned with points of X at which \(\mu\) is discontinuous (i.e. not jointly continuous). The main theorem shows that if X is a connected manifold and \(x_ 0\) in X is an isolated point of discontinuity of \(\mu\) then \(x_ 0\) is the only point discontinuity. If S is taken to be I (the unit interval under min.) then the conclusions of the theorem can be extended to show that \(X.0=\{x_ 0\}\) and that X is homotopy equivalent to a sphere.