Locally contractible space

A locally contractible or locally contractible space is considered in the mathematical sub-area of topology .

definition

A topological space is locally contractible , if for every environment of each point a contractible environment with ${\displaystyle X}$ ${\displaystyle U\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle V\subset X}$

${\displaystyle x\in V\subset U}$

gives.

Examples

Topological ridge

Manifolds and CW-complexes are locally contractible.

Contractible spaces do not always have to be locally contractible. An example of this is the topological ridge , this is defined as

${\displaystyle (\{0\}\times [0,1])\cup (\{1/n|n\in \mathbb {N} \}\times [0,1])\cup ([0,1]\times \{0\})\subset \mathbb {R} ^{2}}$

with the topology induced by the Euclidean metric . ${\displaystyle \mathbb {R} ^{2}}$

Individual evidence

1. ^ John Gilbert Hocking, Gail S. Young: Topology . Dover Publications, New York 1988, ISBN 0-486-65676-4 , pp. 191 .