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Re: Only 45% of the students were prepared for math
Posted:
Mar 28, 2006 2:27 PM
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Michael Stemper wrote (in part):
> I'd gone through chapter 3 of my topology book, which covered
> metric spaces, and was just fine. I could prove things about
> metric spaces, and mappings from one to another. Felt fairly
> confident.
>
>
> I got to chapter 4, which introduced topologies. Unlike the
> previous chapter, there was no motivational material -- just
> the definition of a "topology". I did, indeed, ask myself
> the question of "why arbitrary unions, but only finite
> intersections?"
The following book is excellent for someone who wants to go
from the specific to the general:
Robert Herman Kasriel, "Undergraduate Topology", Krieger
Publishing Company, 1971, xiv + 285 pages.
There's an extensive chapter on topological ideas in R^n:
Euclidean distance properties, open and closed sets,
limit points, continuous functions (and equivalence of
epsilon/delta, sequence, and open set formulations of
continuity), several types of connectedness, compactness
(including limit point compactness, finite open covering
kind, and others), and much more.
Then there's a lengthy chapter that goes through essentially
the same topics for metric spaces, in which some (but not all)
of the earlier equivalences are no longer equivalences.
There's also a lot of motivation for why one would want
to consider the generalization to a metric space, including
applications to existence theorems using fixed points of
contraction operators on certain function spaces.
Finally, at least halfway through the book if not more,
topological spaces are introduced. Again, for the third
time now, essentially the same topics as above are covered,
in which even more of the earlier equivalences wind up
no longer being equivalences.
I think this is an excellent text for someone to read on
their own, especially if they haven't had an advanced
calculus course where much of the R^n material is often
covered. In fact, I read through about the first 85% of
this book as an independent study reading course from
a topologist at a nearby college when I was in high
school. It's probably a bit too repetitious for someone
with a fairly strong background, but for me at the time
and for you (given what you said), I think this book
was/would-be a very good fit.
Dave L. Renfro
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