Alternative non-equivalent definitions[edit]

According to Paul Halmos,^[6] a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest σ-ring containing all compact sets.

Norberg and Vervaat^[7] redefine the Borel algebra of a topological space $X$ as the $\sigma$ -algebra generated by its open subsets and its compact saturated subsets. This definition is well-suited for applications in the case where $X$ is not Hausdorff. It coincides with the usual definition if $X$ is second countable or if every compact saturated subset is closed (which is the case in particular if $X$ is Hausdorff).

Borel hierarchy

Borel isomorphism

Baire set

Cylindrical σ-algebra

– Subfield of mathematical logic

Descriptive set theory

– Concept in topology

Polish space

An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent exposition of Polish topology)

William Arveson

Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989

Richard Dudley

(1950). Measure theory. D. van Nostrand Co. See especially Sect. 51 "Borel sets and Baire sets".

Halmos, Paul R.

Real Analysis, Prentice Hall, 1988

Halsey Royden

Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)

Alexander S. Kechris

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Borel set"

of Borel Sets in the Mizar system, and the list of theorems Archived 2020-06-01 at the Wayback Machine that have been formally proved about it.

Formal definition

"Borel Set". MathWorld.