Definitions of Baire first and second category sets

Question

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!

Answer 1

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.

Answer 2

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.