Let $X$ be a topological space. Then there is a $\sigma$-algebra$\mathcal{B}$ on $X$ generated by the open subsets of $X$ that are preimages of $(0,\infty)$ under some continuous map $X\to\mathbf{R}$. Elements of $\mathcal{B}$ are called the Baire sets (or Baire subsets, or Baire-measurable sets, etc) of $X$, and $\mathcal{B}$ itself is called the Baire $\sigma$-algebra on $X$.