# Why category in Baire's Category Theorem?

1. Jan 1, 2012

### goedelite

Why "category" in Baire's Category Theorem?

I have recently begun reading a monograph on topology, "Introduction to ...." by Gamelin and Greene. Ultimately, I would learn the reason, but patience is not my strong suit. So, why the word "category" in the name of the theorem, which has to do with the density of an intersection of open sets in a complete metric space?

2. Jan 1, 2012

### micromass

Staff Emeritus
Re: Why "category" in Baire's Category Theorem?

The Baire theorem originally dealt with "sets of the first category" and "sets of the second category". Now we call them meagre sets and nonmeagre sets (or fat sets in dutch).

See http://en.wikipedia.org/wiki/Meagre_set

3. Jan 1, 2012

### goedelite

Re: Why "category" in Baire's Category Theorem?

micromass: Thanks for your reply with the citation in wikipedia. I tried reading it and concluded that the import of "category" or "meagre" is a rather detailed matter. That is why, I suppose, it is not explained in my monograph.

I shall have to remain ignorant, allay my impatience by turning my attention elsewhere at least for a while.

4. Jan 6, 2012

### nonequilibrium

Re: Why "category" in Baire's Category Theorem?

goedelite, I think you should be able to understand the first line of the "Definition" section on the wikipedia page linked to above. That's all you need to know to "know" what a meagre subset is.

5. Jan 6, 2012

### goedelite

Re: Why "category" in Baire's Category Theorem?

Mr Vodka: Alas, there is much that I suppose I should be able to do but cannot. That a meagre set is in some sense small or negligible is reassuring in that I could assume there is a categorization of sets: meagre, teeny-weeny bikini, small, medium, large, economy and over-sized.