Equiconnected space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space $X$ is said to be equiconnected if there is a continuous map $k:X\times I\times X\to X$ 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 $y$.

Roughly, speaking, at any given time $t$, we get a map $X\times X$ to $X$. At time $0$, it is projection on the first coordinate, and at time 1, it is projection on the second coordinate. For elements on the diagonal, it always remains the value at the diagonal.