Math 2.0

General topology Definition

Baire property

Builds on Topological spaceOpen setMeagre setSymmetric differenceNowhere dense setInterior

Definition

Let XX be a topological space and let A⊆XA \subseteq X. The set AA has the Baire property if there is an open set U⊆XU \subseteq X such that the symmetric difference A△UA \mathbin{\triangle} U is meagre in XX.

Remarks

Example: every closed set F⊆XF \subseteq X has the Baire property. Take UU to be the interior of FF. Then F△U=F∖UF \mathbin{\triangle} U = F \setminus U is closed and has empty interior, so it is nowhere dense and hence meagre.

Non-example: a Vitali set V⊆ℝV \subseteq \R contains exactly one point of each coset of ℚ\Q in ℝ\R. Such a set does not have the Baire property; its construction uses the axiom of choice.

The sets with the Baire property form a sigma-algebra that contains the open sets, so every Borel set has the Baire property. Equivalently, AA has the Baire property if A=G∪MA = G \cup M for a GδG_\delta set GG and a meagre set MM.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Interior, concept, no evidence levelInteriorconceptMeagre set, concept, no evidence levelMeagre setconceptNowhere dense set, concept, no evidence levelNowhere dense setconceptOpen set, concept, no evidence levelOpen setconceptSymmetric difference, concept, no evidence levelSymmetric differenceconceptTopological space, concept, no evidence levelTopological spaceconceptBaire property, definition, no evidence levelBaire propertydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Interior concept
  2. Meagre set concept
  3. Nowhere dense set concept
  4. Open set concept
  5. Symmetric difference concept
  6. Topological space concept
  7. Baire property definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Property_of_Baire
About this page
TagRR5JH5K74X
URIm2://RR5JH5K74X
Kinddefinition
WritingW0 stub
MSC54E52, 03E15
DefinesBaire property
UsesTopological space, Open set, Meagre set, Symmetric difference, Nowhere dense set, Interior
Changed2 November 2026

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