Equiconnected space

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.

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