A note on -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 . 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) ompact if and only if X is locally compact; s compact, then X is locally connected. 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. o 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.