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The Borel sigma-algebra of a topological space
Definition
Let be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). The Borel sigma-algebra of is the sigma-algebra generated by its open sets:
Its members are the Borel subsets of . The generated sigma-algebra exists and is minimal by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal.
Depends on
Used by
- F-sigma and G-delta subsets of the real line are Borel Example
- The rationals are Borel and F-sigma but neither open nor closed nor G-delta Example
- A continuous map has Borel preimages of Borel sets Theorem
- For n at least one, open sets, closed sets, compact sets, open balls, boxes, rational open boxes, and rational half-open boxes generate the Borel sigma-algebra on Rⁿ Theorem
- Seven generating families for the Borel sigma-algebra on the real line Theorem
- The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 6 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- T. Tao, An Introduction to Measure Theory, Definition 1.4.16 (standard reference, not scraped)
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Section 2.1 (standard reference, not scraped)