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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.

Any point of $\mathbb{R}$ 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].

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
Canonical name	Cutpoint
Date of creation	2013-03-22 13:56:38
Last modified on	2013-03-22 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