Topological space , containing a point whose closure is the whole space, is contractible?

Question

Let $X$ be a topological space containing a point whose closure is the whole space $X$. Then is $X$ contractible ?

I feel it is, but I am unable to come up with a proof. Please help.

Answer 1

If $\eta$ is the dense point and $I=[0,1]$ the map $f:X\times I\to X$ defined by $$f(x,0)=x \quad \operatorname {and} \quad f(x,t)=\eta \quad \operatorname {for} t\gt0$$ is the required contraction of $X$ to $\eta$ .