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From Planetmath

A meager or Baire first category set in a topological space is one which is a countable union of nowhere dense sets.

A Baire second category set is one which contains a countable union of open and dense sets.

From Wikipedia:

A subset of a topological space X is called

  • nowhere dense in X if the interior of its closure is empty
  • of first category or meagre in X if it is a union of countably many nowhere dense subsets
  • of second category or nonmeagre in X if it is not of first category in X

I was wondering

  1. according to Wikipedia's definition, is any subset of a topological space either of first category or of second category?
  2. are the definitions for second category set in Planetmath and Wikipedia consistent with each other?
  3. Wikipedia says these definitions are used for "historical definition" of Baire space. I was wondering if they are archaic i.e. no longer in use?

Thanks and regards!

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What is your doubt in answering question 1? –  wildildildlife Feb 6 '12 at 13:42
@wildildildlife: no now. I felt a little unsure. –  Tim Feb 6 '12 at 13:49
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2 Answers

up vote 2 down vote accepted

a. It is true you are either a countable union of nowhere dense sets or you are not. Thus, any set is is either of first category or second category.

b. It is stated in the wrong way. It should have said "contains a countable intersection of open dense sets", not union. Note that the complement of an open dense set is closed nowhere dense (and vice versa).

c. The notation is still in use.

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In addition to the correction you made in (b), the Planetmath definition seems to be conflating "$A$ is second category" and "$A$'s complement is first category." –  user83827 Feb 5 '12 at 1:37
@ccc: I was wondering if "A is second category" and " A's complement is first category" are equivalent? –  Tim Feb 5 '12 at 2:31
No, there are in general plenty of second category sets with second category complement. For example, $(0,1)$ as a subset of the reals. –  user83827 Feb 5 '12 at 2:38
Indeed. The fact that a set of reals can be nonmeagre and have a nonmeagre complement is part of the reason why I think the meagre/nonmeagre/comeagre terminology should be preferred. In addition it's more "natural" (in terms of language): "X is comeagre" sounds much nicer than "X is the complement of a set of the 1st category". –  kahen Feb 5 '12 at 6:12
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so to recap: the more modern definition

1) $X$ is a Baire space iff every countable union of closed nowhere dense sets has empty interior.

Equivalent by taking complements (note that a set $A$ is nowhere dense iff its complement $X \setminus A$ contains an open dense subset) to my favourite formulation, which seems to be more commonly used among topologists:

1') $X$ is a Baire space iff every countable intersection of open and dense subsets is dense.

(note that in any space a finite intersection of open and dense subsets is open and dense, so the countable intersection is the first "interesting" question, in a way.)

And what they call the historical definition:

2) Every non-empty open subset of $X$ is of second category.

The article call it historical because it uses a notion "category" of a subset (a subset is either first category or second category, and not both, by definition), which has fallen in some disuse. Nowhere dense sets and meagre sets (the countable unions of nowhere dense subsets) are still normal usage. Note that a first category subset is now called meagre, and the notion of "second category" is not used as much (but still occurs), so it's good to know it. But definition 1) and 2) are easily proved to be equivalent, so they give rise to the same spaces being called Baire. So we have a trivial reformulation of the "classical" definition 2 as:

2') Every non-empty open subset of $X$ is non-meagre.

Or, stated more "positively"

2'') Every meagre set has empty interior.

(otherwise the non-empty interior is a subset of a meagre set, and thus meagre etc.)

which brings us back to definition 1) again.

It's just that the Wikipedians do not like the category terminology (because it might confuse people with category theory as a branch of maths) and so choose to reformulate everything using meagre and non-meagre instead.

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Regardless of possible confusion with category theory (which I think is neglible), I find it a bad naming: 'of first category' is totally undescriptive ('nowhere dense' and 'meagre' do not suffer from this). In the same spirit, I also dislike 'Type X' in the theory of Von Neumann Algebras, and '$T_i$' indicating a certain separation axiom. –  wildildildlife Feb 6 '12 at 13:39
@wildildildlife Agreed, it's non-descriptive, as is "normal" (for any property, really). The $T_i$ at least has the advantage of being a hierarchy, in the sense that the higher ones imply the lower ones, at least in the definitions I was taught, where $T_4$ = normal + $T_1$ etc. –  Henno Brandsma Feb 6 '12 at 17:44
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