Explain what is meant by a Hausdorff (topological) space, and prove that every compact subset of a Hausdorff space is closed.

Let $X$ be an uncountable set, and consider the topology $\mathcal{T}$ on $X$ defined by

$$U \in \mathcal{T} \Leftrightarrow \text { either } U=\emptyset \text { or } X \backslash U \text { is countable. }$$

Is $(X, \mathcal{T})$ Hausdorff? Is every compact subset of $X$ closed? Justify your answers.