View source for Baire property



''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150401.png" /> in a topological space''

A property analogous to the measurability of a set. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150402.png" /> has the Baire property if there exists an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150403.png" /> such that the differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150405.png" /> are sets of the first category according to Baire (cf. [[Category of a set|Category of a set]]); the term  "open"  may be replaced by  "closed" . There are other equivalent definitions; thus, for instance, a set has the Baire property if it is the union of a set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015040/b0150406.png" /> and a set of the first category. The class of sets with the Baire property is closed under the operations of complementing, taking countable unions and taking countable intersections. For an example of a set which does not have the Baire property, see [[#References|[1]]].

====Comments====
A set with the Baire property is often called a Baire set or an almost-open set. A set of the first category is often called meager.

====References====
{|
|valign="top"|{{Ref|Ba}}||  R. Baire,   "Leçons sur les fonctions discontinues, professées au  collège de France" , Gauthier-Villars  (1905) {{ZBL|36.0438.01}}
|-
|valign="top"|{{Ref|Ku}}|| K. Kuratowski,   "Topology" , '''1–2''' , Acad. Press  (1966–1968) {{MR|0259835}}  {{MR|0217751}} {{ZBL|0158.40802}} {{ZBL|0148.42801}}  {{ZBL|0104.27402}} {{ZBL|0098.24110}} {{ZBL|0137.15604}}  {{ZBL|0041.09604}} {{ZBL|0041.09603}} {{ZBL|0008.13202}}   {{ZBL|56.0517.04}}
|-
|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,   "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL| 0217.09201}}
|-
|valign="top"|{{Ref|Ro}}||  H.L. Royden, "Real analysis", Macmillan  (1968) {{MR|0151555}} {{ZBL|0197.03501}}
|-
|}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,   "Topological spaces" , Wiley  (1966)  {{MR|0211373}} {{ZBL|0141.39401}} </TD></TR></table>