cut-point

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

0.0.1 Examples

1. Any point of $ℝ$ with the usual topology is a cut-point.

2. If $X$ is a normed vector space with $dimX>1$, then $X$ hasno cut-points [1].