The Stone-Čech compactification of an irreducibly connected space
Glenn L. Pfeifer · Proceedings of the American Mathematical Society · 1969
A connected topological space X is irreducibly connected about a subset A EX (written X is an A-i-connex) if no proper connected subspace of X contains A. A connected space Y is irreducibly closed connected about a subset BEY (written F is a F-î-C-connex)if no proper closed connected subspace of F contains B. The structure of such spaces has been studied by Gehman [l], Wilder [7] and Strebe [3], [4], among others.In this paper we show that the Stone-Cech compactificationßX of an î'-connex X is an j-connex, and that ß Y, for F a suitably restricted i-C-connex, is an i-C-connex.The author would like to express his appreciation to Professor Edwin Halfar for many conversations concerning this paper.2. Notation.All spaces are at least Tx and completely regular.For SCX, Cl 5 is the closure of 5 in X, and for TEßX, Clfl F is the closure of F in ßX.If SCX, then S°=ßX-Clß (X-S).