A-Sets and Abcohesive Spaces
David John · Missouri Journal of Mathematical Sciences · 1993
Definitions.A space M is abcohesive at a point p with respect to a point q if there exists an open connected set U such that p is a point in U and U is a subset of M -{q}.The space M is abcohesive at a point p if it is abcohesive at p with respect to q for each q in M -{p}.The space M is abcohesive if it is abcohesive at p for each p in M .Remarks.If p is a non-cut point of M , and M is T 1 then M is abcohesive at each point q in M -{p} with respect to p. Hence, if each point of M is a non-cut point of M , then M is abcohesive.Also, if M is a locally connected T 1 space, then M is abcohesive.Sierpinski space is locally connected but not abcohesive.However, Sierpinski space is not T 1 .For the remainder of this paper, we will assume the space M is Hausdorff.If M is a continuum, then there exist two points p and q in M such that M is abcohesive at each x in M -{p} with respect to p and at each x in M -{q} with respect to q.