Date: Thu, 19 Oct 1995 12:40:58 -0700
From: [Permission pending]
To: rusin@washington.math.niu.edu
Subject: Re: What's the homology of this space?

In article <46637q$q5k@muir.math.niu.edu> you write:
>Let  X  be the subspace of  R^2  obtained by deleting the points
>with both coordinates rational.
>
>What is  H_1(X,Z)?  What is  H^1(X,Z)?
>
>I would be interested in pointers to the literature. I hope not to
>have to get into  lim^1  issues and so on, but that may be inevitable.
>
>dave
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I would bet that the answer for such a "pathological" space will depend
strongly on exactly which (co)homology theory you are using: 
simplicial, singular, compact support, Cech, etc.

(Incidentally, as you may know, your space X is homeomorphic to
R^2 - D, where D is any countable dense subset of R^2.)

--[Permission pending]

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Date: Fri, 20 Oct 1995 14:44:20 +0100
From: David Epstein <mareb@csv.warwick.ac.uk>
To: rusin@washington.math.niu.edu
Subject: Re: What's the homology of this space?

With reference to your article in sci.math.research,
I think the answer must depend on which homology theory you use.

For example, if you use Cech homology, then this is based on open
coverings. Every open covering of your space has  a countable
subcovering. Therefore the Cech homology is a direct limit of
countable groups and there's a good chance that the answer is
countable (I'm not sure of this).

If you use singular homology, then you have phenomena like the Hawaian
earring. This presumably ensures that the group is uncountable.

And there are other homology and cohomology theories. When dealing
with a horrible space like the one you mention, it's important to fix
on one particular theory.

Best wishes,

David Epstein.