A space X is said to have the fixed point property if every map $f: X \to X$ has a fixed point, i.e. a point $x_0 \in X$ such that $f(x_0) = x_0$. Prove that a space having the fixed point property must be connected.
completely stuck on it.can anyone help me please.thanks for your time
Suppose that $X$ can be written as the disjoint union of two open sets $O_1$ and $O_2$. Let $x_1 \in O_1$ and $x_2 \in O_2$. You can show that if $f : X \to X$ is such that $f(x)=x_1$ if $x \in O_2$ and $f(x)=x_2$ if $x \in O_1$, then $f$ is continuous and does not have any fixed point.
