cutpoint
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 cutpoint of $X$ [1, 2].
0.0.1 Examples

1.
Any point of $\mathbb{R}$ with the usual topology is a cutpoint.

2.
If $X$ is a normed vector space^{} with $dimX>1$, then $X$ has no cutpoints [1].
References
 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.
Title  cutpoint 

Canonical name  Cutpoint 
Date of creation  20130322 13:56:38 
Last modified on  20130322 13:56:38 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  5 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 54D05 
Synonym  cutpoint 