#definition **Def.** \A [**Borel set**](https://en.wikipedia.org/wiki/Borel_set#Standard_Borel_spaces_and_Kuratowski_theorems) is any set in a [topological space](https://en.wikipedia.org/wiki/Topological_space "Topological space") that can be formed from [open sets](https://en.wikipedia.org/wiki/Open_set "Open set") (or, equivalently, from [closed sets](https://en.wikipedia.org/wiki/Closed_set "Closed set")) through the operations of [countable](https://en.wikipedia.org/wiki/Countable "Countable") [union](https://en.wikipedia.org/wiki/Union_(set_theory) "Union (set theory)"), countable [intersection](https://en.wikipedia.org/wiki/Intersection_(set_theory) "Intersection (set theory)", and [relative complement](https://en.wikipedia.org/wiki/Relative_complement "Relative complement"). **Fact.** For a topological space _X_, the collection of all Borel sets on _X_ forms a [σ-algebra](https://en.wikipedia.org/wiki/Sigma-algebra "Sigma-algebra"), known as the **Borel algebra** or **Borel σ-algebra**. The Borel algebra on _X_ is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). **Def.** Let _X_ be a topological space. The **Borel space** associated to _X_ is the pair (_X_,_B_), where _B_ is the σ-algebra of Borel sets of _X_. **Def.** A **Polish space** is a [separable](https://en.wikipedia.org/wiki/Separable_space "Separable space") [completely metrizable](https://en.wikipedia.org/wiki/Completely_metrizable_space "Completely metrizable space") [topological space](https://en.wikipedia.org/wiki/Topological_space "Topological space"); that is, a space [homeomorphic](https://en.wikipedia.org/wiki/Homeomorphic "Homeomorphic") to a [complete](https://en.wikipedia.org/wiki/Complete_space "Complete space") [metric space](https://en.wikipedia.org/wiki/Metric_space "Metric space") that has a [countable](https://en.wikipedia.org/wiki/Countable "Countable") [dense](https://en.wikipedia.org/wiki/Dense_set "Dense set") subset. **Def.** A **standard Borel space** is the Borel space associated to a Polish space. **Thm. (Kuratowski)** Theorem. Let X be a Polish space. Then X as a Borel space is Borel isomorphic to one of (1) $\mathbb {R}$ (2) $\mathbb {Z}$ or (3) a finite space.