Is the Baire space $\sigma$-compact?
The Baire space is the set $\mathbb{N}^\mathbb{N}$ of all sequences of natural numbers under the product topology taking $\mathbb{N}$ to be discrete. It is a complete metric space, for example with the metric $d ( x , y ) = \frac{1}{n+1}$ where $n$ is least such that $x(n) \neq y(n)$.
A topological space $X$ is called $\sigma$-compact if it is the countable union of compact subsets.