Topological ridge

Topological ridge

The topological ridge is a topological space that is a source of counterexamples in topology .

definition

The topological ridge is given by

${\displaystyle K=(\{{\tfrac {1}{n}}|n\in \mathbb {N} \}\times [0,1])\cup ([0,1]\times \{0\})\cup (\{0\}\times [0,1])\subseteq \mathbb {R} ^{2}}$

equipped with the subspace topology.

The deleted topological ridge is given as . ${\displaystyle D=K\setminus (\{0\}\times (0,1))}$

Topological properties

The topological ridge is an example of a topological space that is path-connected , but not locally path-connected , because no point from has a neighborhood basis made of path-connected sets. ${\displaystyle \{0\}\times (0,1]}$

The deleted topological ridge is an example of a topological space that is connected but not path-connected because there is no path between and . ${\displaystyle (0,0)}$ ${\displaystyle (0,1)}$

swell

• James Munkres : Topology , 2nd edition, Prentice Hall , 1999, ISBN 0-13-181629-2 .