Theorem Suppose $X$ is a connected space and $x$ is a point in $X$ . If $X\setminus \{x\}$ is a disconnected set in $X$ , then $x$ is a cut-point of $X$ [1,2].

Examples

1. Any point of $\sR$ with the usual topology is a cut-point.
2. If $X$ is a normed vector space with $\dim X>1$ , then $X$ has no cut-points [1].

Bibliography

1

G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.

2

L.E. Ward, Topology, An Outline for a First Course, Marcel Dekker, Inc., 1972.