Bcd at 03:51, 20 December 2010

2010-12-20T03:51:25Z

← Older revision		Revision as of 23:51, 19 December 2010
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	Or maybe, for the entire subject--math, if you believe it then it is right. If you don't believe it then it		Or maybe, for the entire subject--math, if you believe it then it is right. If you don't believe it then it
	is completely wrong.-Kai		is completely wrong.-Kai




			Too late for marks, but just in time for the holidays are the extra problems (1,6,8,12 pp 170-171) - just sketches:

			1. (a) is trivial. For (b), suppose X is compact Hausdorff relative to topologies <math>T_1, T_2</math> which are comparable. The identity function Id is continuous in one direction and so takes compact sets to compact sets. The rest is easy: fix any open set U and consider Id(X-U) ...

			6. Also trivial. As in 1(b), use (i) compact subsets of T2 spaces are closed and (ii) closed subsets of compact spaces are compact.

			8. If f is continuous, fix any point (x,y) not on the graph. Separate y and f(x) by open sets U, V, respectively. By continuity, find W open in X st f(W) is a subset of V. Then W x U is open disjoint from the graph. Conversely, if the graph is closed, apply the hint ...

			12. Fix an open cover <math>U_\alpha</math> for X. For each y in Y, its preimage under p is compact, so can be covered by some finite collection of the <math>U_\alpha</math>, which also gives an open set. Do this for every y in Y. But then apply the hint to notice that we get a covering of Y by open neighbourhoods; apply compactness and the hint to conclude we only care about the preimages of finitely many of these nbds in Y, which are already inside finitely many finite unions of open sets.

			[[User:Bcd\|Bcd]] 22:51, 19 December 2010 (EST)

Anne.d at 02:54, 11 December 2010

2010-12-11T02:54:36Z

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===Solution===
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An assignment without a solution is like a nightmare to me. I like every question accompanied
with a clean solution aside.(Might not be the case for research question because they are just
simply too hard.) I would like to share this happiness of understanding and acquiring knowledge
with everybody because I don't think this class is a battle. I certainly don't like the idea that
we should keep information/answers as something like business secrets. This learning process
should be enjoyable which should be full of discussions instead of things like "you have to think on
your own/ I can't tell you the answer". I know maybe other people might not agree with me but I believe
a positive learning environment is crucial to truly understanding something well although we should not neglect
independent thinking at the same time. That is why I share whatever I have with you. If I am wrong feel free to criticize
me, and I am pretty sure a lot of people don't agree with me. But that is OK because there is just no
absolute right or wrong and everybody is doing what they think is right. Just like you can't say if
Axiom of choice is right or not. If you believe then it is right. If you don't believe it then it is wrong.
Or maybe, for the entire subject--math, if you believe it then it is right. If you don't believe it then it
is completely wrong.-Kai

10-327/Homework Assignment 5 Solutions - Revision history

Bcd at 03:51, 20 December 2010

Anne.d at 02:54, 11 December 2010