Explicit homotopy equivalence

Question

Assume $K\subseteq X$ is a non-empty contractible subspace of a topological space $X$. Let $x\in K$ be any point. I am trying to define an explicit homotopy equivalence between $X \setminus K$ and $X\setminus \{x\}$. How would I go about doing this? Thanks for your help^^

Answer 1

This is false. Let $X = [0,1], K = [0,1/2]$ and $x = 1/2$.