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Re: Only 45% of the students were prepared for math
Posted:
Mar 30, 2006 11:56 AM
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In article <200603281743.k2SHhi9129832@walkabout.empros.com>,
Michael Stemper <mstemper@siemens-emis.com> wrote:
>[newsgroups trimmed]
>In article <e02vhb$q1a$1@news.math.niu.edu>, Dave Rusin writes:
>>In article <e01oge$4u9$1@glue.ucr.edu>, James Dolan <jdolan@math-cl-n03.math.ucr.edu> wrote:
>>>in article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>, the world wide wade <waderameyxiii@comcast.remove13.net> wrote:
>>>|And if you haven't seen metric
>>>|spaces, the axioms for point-set topology will appear like some
>>>|arbitrary abstract nonsense from another planet.
>>>grossly false;
>>OK, I have a bright student uncorrupted by point-set topology before me.
>>I present the definition of a topology; the collection of open sets
>>is closed under _finite_ intersections but _arbitrary_ unions. The
>>student asks why on earth one would take such an asymmetrical set of
>>axioms.
>Yup, that'd be me.
>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".
There should have been an explanation of why. Moreover, you
expected it to be like metric spaces, where in general the
metric is rather arbitrary form the topological standpoint,
and convergence definitions are ill-suited for extension.
I did, indeed, ask myself the question of "why
>arbitrary unions, but only finite intersections?"
Were you not aware that in a metric space any closed set is
an intersection of a sequence of open sets? Or at least any
closed interval in the real line is such an intersection?
And that the metric definition of open set gets arbitrary
unions, but only finite intersections in general?
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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