Let $T:X\to X$ be a continuous map defined on the compact space $X$. I read that if $X$ is a metric space then the set of T-invariant probability measure is not empty.
I want to know if this result is true when $X$ is only a compact topological space (maybe Hausdorff if it's neccesary). I tried to construct a counterexample but I couldn't =(
The proof that I read uses some properties of metric space so I could not reply the metric case proof =(