Equiconnected implies contractible

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., equiconnected space) must also satisfy the second topological space property (i.e., contractible space)
View all topological space property implications | View all topological space property non-implications
Get more facts about equiconnected space|Get more facts about contractible space

Statement

Any equiconnected space is contractible.

Definitions used

Equiconnected space

Further information: equiconnected space

A nonempty topological space $X$ is said to be equiconnected if there is a continuous map $k:X\times [0,1]\times X\to X$ such that $k(x,t,x)=x$ for all $x$ and $k(x,0,y)=x,k(x,1,y)=y$ for all $x$ and $y$.

Contractible space

Further information: contractible space

A nonempty topological space $X$ is said to be contractible if there exists a point $x_0\in X$ and a continuous map $F:X\times [0,1]\to X$ such that $F(x,0)=x$ for all $x$, and $F(x,1)=x_0$ for all $x$. (Another equivalent definition states that this is true for every $x_0\in X$).