A necessary and sufficient condition for 𝛽𝑋\𝑋 to be an indecomposable continuum

Abstract

In his dissertation, David Bellamy has shown that if I = [ 0 , 1 ) I = [0,1) , then β I ∖ I \beta I\backslash I is an indecomposable continuum, and R. G. Woods, in his dissertation, obtained the same result and in addition showed that for m > 1 , β R m ∖ R m m > 1,\beta {R^m}\backslash {R^m} is a decomposable continuum. In this note we give a necessary and sufficient condition for β X ∖ X \beta X\backslash X to be an indecomposable continuum when X is a locally connected generalized continuum.