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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeJun 24th 2011
On the page Baire space, I see a comment that the Baire space of irrationals is "not a very important example". I think it actually is a pretty important example; for people working with Polish spaces and descriptive set theory, it is often used as an archetypal example. At least I don't understand what the purpose of this parenthetical remark would be.
• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeJun 27th 2011
• (edited Jun 27th 2011)

It’s a very important space to some people, which is why we (I) write about it (at its own page). But is it important as a Baire space? (For example, if every quotient space of a Baire space were Baire, then this would be important, and it would follow that every Polish space is Baire.)

In other words, the Baire category theorem for complete metric spaces is important, and the Baire category theorem for locally compact Hausdorff spaces is important, but is the Baire category theorem for $J$ important? I don’t think so, but maybe I’m wrong about this.

Another issue is that this space is an example of a complete metric space, so why bring it up specifically in the list of examples? Only because of the coincidence of the names. (And it is a coincidence, as far as I can tell, although I would be delighted to learn otherwise.)

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJun 27th 2011
• (edited Jun 27th 2011)

I still don’t understand the need for the parenthetical comment, because it’s no less important than any other example (and it’s plenty important in other contexts). It just seems like the comment might lead to some confusion in people’s minds. I think it’s enough just to disambiguate the terminology.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJun 27th 2011

Maybe I’m making too big a deal about this, but I went ahead and reworded it to reflect what I think what was meant. (But I’m not as fussed about it as I was earlier.)

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeJun 30th 2011

OK. I promoted it again to an actual example, which I assume you’re happy with.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 28th 2017

In order to un-gray links, I gave Baire category theorem an entry, but at the moment it does nothing but point to Wikipedia.

The lead-in sentence of the Idea-section at Baire space is really not helpful, it should be changed. Instead I added pointer to “Baire category theorem” in the Examples-section, where the theorem was stated without naming it.