is there any contractable space with three points?

Question

i'm looking for a three point contractible topological space, is this example true? T={{(0,1]},{(1,2]},{(0,2]},{(0,3]},empty set}

Answer 1

Your $T$ mentions (at least) four points and doesn't define a topology.

Here's a hint. Every indiscrete space is contractible; indeed, if $X$ is indiscrete, then any function $H : X \times [0,1] \to X$ is a homotopy since every function whose codomain is indiscrete is continuous.