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

作者：Peng, Liang-Xue^*; Cao, Shi-Tong

来源：Topology and Its Applications, 2013, 160(5): 739-747.

DOI：10.1016/j.topol.2013.01.023

• Connected space
• Cut point
• cut*-space
• cut((n))-space
• H(i)
• H-set

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 vertical bar DI vertical bar = 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 subset of X with vertical bar D vertical bar >= 3 such that X\D is disconnected and X\C is connected for every proper subset C subset of D. We give some sufficient conditions that the set {x} is open or closed if x is an element of 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.

• 出版日期2013-3-15
• 单位北京工业大学