Date: Tue, 21 Feb 1995 17:51:56 +0500
From: gordon@atria.com (Gordon McLean Jr.)
To: rusin@math.niu.edu
Subject: Re: Metrization (Was: Re: Non-standard analysis)

> From rusin@math.niu.edu Fri Feb 17 13:14 EST 1995
> Date: Fri, 17 Feb 95 12:13:36 CST
> From: rusin@math.niu.edu (Dave Rusin)
> To: gordon@atria.com
> Subject: Re: Metrization (Was: Re: Non-standard analysis)
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> 
> This is just opinion of course but:
> 
> Paracompactness is an internal property of topological spaces (unlike
> having a metric, which requires the presentation of an additional structure).
> However, it is, as you have noted, as much like being a metric space as
> you can expect of an internal condition which is, really, not that hard
> to state. So I'd have to say paracompactness is pretty important if you
> want strong theorems which will, of course, only hold for "nice" spaces
> (that's the kind of theorem I prefer, over weak-but-general theorems).
> 
> This is particularly helpful in differential topology and differential
> geometry; the issue there is the ability to create "partitions of unity",
> that is, collections of functions which add up to  1  at every point,
> each one of which however is zero except in a small region. Multiplying
> these functions by any other function makes it possible to study all other
> functions locally. On a manifold, that reduces everything to functions on
> R^n, which is calculus and related fields. So that's what paracompactness
> does: it allows a passage from local to global.

Good point.

> You've already noted that
> in the metrization theorem you quoted.
> 
> On the other hand, paracompactness is not of much help in algebraic topology,
> except perhaps for preparing other conditions which in turn are more
> suited for algebraic topology. Algebraic topology may be thought of as a
> method for computing how a space is built up from simpler pieces, where
> "simple" is usually measured relative to some well-known subsets of
> Euclidean space (spheres, simplices, intervals,...) As such it's no big
> deal to assume from the start that your space has some additional
> structure anyway -- CW complex, PL manifold, or whatever. In that case,
> you can go ahead and assume you've got a metric space, say, and not
> bother with paracompactness.

Another good point.  Munkres's own algebraic topology book never
references paracompactness, as I recall.  (In fairness though, his
development of homology theory requires surprisingly little non-elementary
point set topology, as he himself notes in the preface.)

> 
> I've heard Munkres' opinion of paracompactness before, and I see what he
> has in mind, but I think topology is a big enough field that there's
> plenty of room for life without it.

Thank you for this thoughtful response.

Regards,

Gordon McLean, Jr.