# cut-point

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].

## 0.0.1 Examples

1. 1.

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

2. 2.

If $X$ is a normed vector space with $\dim X>1$, then $X$ has no cut-points [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 cut-point Cutpoint 2013-03-22 13:56:38 2013-03-22 13:56:38 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 54D05 cutpoint