A note on cut(n)-spaces and related conclusions

A connected topological space X is said to be a cut(n)-space for some natural number n, if X\D is disconnected for any subset D of X with |D|=n and X\Y is connected for each proper subset Y of D. A cut(n)-space is also called a cut point space if n=1 and a cut⁎-space if n=2.We get the following conclusions: If n⩾2, then a cut(n)-space is a Hausdorff space. If X is a cut(2)-space, then the following statements hold:(1)X is compact if and only if X is locally compact;(2)If X is compact, then X is locally connected. If X is a locally compact topological space, then X is not a cut(n)-space for each n⩾3. We point out that there exists a topological space X with a finite set D⊂X with |D|⩾3 such that X\D is disconnected and X\C is connected for every proper subset C⊂D. We give some sufficient conditions that the set {x} is open or closed if x∈D and the set D has the above properties.We also discuss a property on H-sets of a connected topological space and discuss some properties of H(i) topological spaces. Finally, we show that in a COTS the closure of each cut point contains at most three points and in a connected space with endpoints the closure of each endpoint contains at most one point other than the endpoint.