Editing Borel set (section)

==Alternative non-equivalent definitions==

According to [[Paul Halmos]],<ref>{{harv|Halmos|1950|loc=page 219}}</ref> a subset of a [[locally compact]] Hausdorff topological space is called a ''Borel set'' if it belongs to the smallest [[Sigma-ring|σ-ring]] containing all compact sets.

Norberg and Vervaat<ref>Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: ''Probability and Lattices'', in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150</ref> redefine the Borel algebra of a topological space <math>X</math> as the <math>\sigma</math>-algebra generated by its open subsets and its compact [[Saturated set|saturated subset]]s. This definition is well-suited for applications in the case where <math>X</math> is not Hausdorff. It coincides with the usual definition if <math>X</math> is [[second countable]] or if every compact saturated subset is closed (which is the case in particular if <math>X</math> is Hausdorff).