Editing Meagre set (section)

==Definitions==

Throughout, <math>X</math> will be a [[topological space]]. 

The definition of meagre set uses the notion of a [[nowhere dense]] subset of <math>X,</math> that is, a subset of <math>X</math> whose [[closure (topology)|closure]] has empty [[interior (topology)|interior]].  See the corresponding article for more details.

A subset of <math>X</math> is called '''{{visible anchor|meagre in}} <math>X,</math>''' a '''{{visible anchor|meagre subset|text=meagre subset}} of <math>X,</math>''' or of the '''{{visible anchor|first category}} in <math>X</math>''' if it is a countable union of [[nowhere dense]] subsets of <math>X</math>.{{sfn|Narici|Beckenstein|2011|p=389}} Otherwise, the subset is called '''{{visible anchor|nonmeagre in}} <math>X,</math>''' a '''{{visible anchor|nonmeagre subset|text=nonmeagre subset}} of <math>X,</math>''' or of the '''{{visible anchor|second category}} in <math>X.</math>'''{{sfn|Narici|Beckenstein|2011|p=389}}  The qualifier "in <math>X</math>" can be omitted if the ambient space is fixed and understood from context.

A topological space is called '''{{visible anchor|meagre|meagre space}}''' (respectively, '''{{visible anchor|nonmeagre|nonmeagre space}}''') if it is a meagre (respectively, nonmeagre) subset of itself. 

A subset <math>A</math> of <math>X</math> is called '''{{visible anchor|comeagre}} in <math>X,</math>''' or '''{{visible anchor|residual|residual subset|residual set}} in <math>X,</math>''' if its [[complement (set theory)|complement]] <math>X \setminus A</math> is meagre in <math>X</math>. (This use of the prefix "co" is consistent with its use in other terms such as "[[Cofiniteness|cofinite]]".)
A subset is comeagre in <math>X</math> if and only if it is equal to a countable [[intersection (set theory)|intersection]] of sets, each of whose interior is dense in <math>X.</math> 

'''Remarks on terminology'''

The notions of nonmeagre and comeagre should not be confused.  If the space <math>X</math> is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets.  If the space <math>X</math> is nonmeager, no set is at the same time meagre and comeager, every comeagre set is nonmeagre, and there can be nonmeagre sets that are not comeagre, that is, with nonmeagre complement.  See the Examples section below.

As an additional point of terminology, if a subset <math>A</math> of a topological space <math>X</math> is given the [[subspace topology]] induced from <math>X</math>, one can talk about it being a meagre space, namely being a meagre subset of itself (when considered as a topological space in its own right).  In this case <math>A</math> can also be called a ''meagre subspace'' of <math>X</math>, meaning a meagre space when given the subspace topology.  Importantly, this is not the same as being meagre in the whole space <math>X</math>.  (See the Properties and Examples sections below for the relationship between the two.)  Similarly, a ''nonmeagre subspace'' will be a set that is nonmeagre in itself, which is not the same as being nonmeagre in the whole space.  Be aware however that in the context of [[topological vector spaces]] some authors may use the phrase "meagre/nonmeagre subspace" to mean a vector subspace that is a meagre/nonmeagre set relative to the whole space.<ref>{{cite web |last1=Schaefer |first1=Helmut H. |title=Topological Vector Spaces |url=https://books.google.com/books?id=5_1QAAAAMAAJ&q=%22meager+subspace%22 |publisher=Macmillan |date=1966}}</ref>

The terms ''first category'' and ''second category'' were the original ones used by [[René Baire]] in his thesis of 1899.<ref>{{cite journal |last1=Baire |first1=René |title=Sur les fonctions de variables réelles |journal=Annali di Mat. Pura ed Appl. |date=1899 |pages=1–123 |url=https://archive.org/details/surlesfonctions00bairgoog/page/n12/mode/2up |series=3}}, page 65</ref>  The ''meagre'' terminology was introduced by [[Nicolas Bourbaki|Bourbaki]] in 1948.<ref>{{cite journal |last1=Oxtoby |first1=J. |title=Cartesian products of Baire spaces |journal=[[Fundamenta Mathematicae]] |date=1961 |volume=49 |issue=2 |pages=157–166 |doi=10.4064/fm-49-2-157-166 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49113.pdf}}"Following Bourbaki [...], a topological space is called a Baire space if ..."</ref>{{sfn|Bourbaki|1989|p=192}}