10-327/Homework Assignment 8 Solutions

DBN: Classes: 2010-11: 10-327/Navigation Additions to the MAT 327 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 13 About This Class, Monday - Continuity and open sets, Thursday - topologies, continuity, bases. 2 Sep 20 Monday - More on bases, Thursdsay - Products, Subspaces, Closed sets, HW1, HW1 Solutions 3 Sep 27 Monday - the Cantor set, closures, Thursday, Class Photo, HW2, HW2 Solutions 4 Oct 4 Monday - the axiom of choice and infinite product spaces, Thursday - the box and the product topologies, metric spaces, HW3, HW3 Solutions 5 Oct 11 Monday is Thanksgiving. Thursday - metric spaces, sequencial closures, various products. Final exam's date announced on Friday. 6 Oct 18 Monday - connectedness in ${\mathbb {R} }$, HW4, HW4 Solutions, Thursday - connectedness, path-connectedness and products 7 Oct 25 Monday - Compactness of $[0,1]$, Term Test on Thursday, TT Solutions 8 Nov 1 Monday - compact is closed and bounded, maximal values, HW5, HW5 Solutions, Wednesday was the last date to drop this course, Thursday - compactness of products and in metric spaces, the FIP 9 Nov 8 Monday-Tuesday is Fall Break, Thursday - Tychonoff and a taste of Stone-Cech, HW6, HW6 Solutions 10 Nov 15 Monday - generalized limits, Thursday - Normal spaces and Urysohn's lemma, HW7, HW7 Solutions 11 Nov 22 Monday - $T_{3.5}$ and $I^{A}$, Thursday - Tietze's theorem 12 Nov 29 Monday - compactness in metric spaces, HW8, HW8 Solutions, Thursday - completeness and compactness 13 Dec 6 Monday - Baire spaces and no-where differentiable functions, Wednesday - Hilbert's 13th problem; also see December 2010 Schedule R Dec 13 See December 2010 Schedule F Dec 20 Final exam, Monday December 20, 2PM-5PM, at BR200 Register of Good Deeds Add your name / see who's in! See Hilbert's 13th

In order to make our topological space of homework assignments complete, I need to upload the solution to the last assignment.(Sorry for getting so busy and keep forgetting things) What metric we should put on it?...Its quite difficult to create a complete space...

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-Kai Xwbdsb 07:32, 19 December 2010 (EST)

I think the set of homeworks is a finite point set, hence automatically compact ... if you view the homeworks as spaces instead of points, then the resulting product space is complete if each homework is complete, as we know ... exercise: (a) is the converse also true? (b) what about infinite products?

-The resulting product space is complete only if you put that weird metric that induces the product topology. When the space is not metric we cannot talk about completeness. Converse depends on what kind of metric you define on the projection? countable infinite product is done in the book. It is complete with that weird metric induced from metrics from all the spaces.-Kai

Anyway, here is another solution set, this one typed (though not graded, so use at your own risk): Bcd 14:40, 19 December 2010 (EST)

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