Is the Open Box Open?

Tuesday, July 24th, 2012

A quick proof of something that bothered me in basic topology. Assume the standard topology on â„n based on open balls. What about an open box? I.e. all points in â„n such that a1 < x1 < b1; a2 < x2 < b2;…;an < xn < bn. Is this an open set? I.e. can you build it up out of a union of open balls? Or, more colloquially, can you pack a square hole with round pegs without leaving any gaps?

Short answer: yes, if the balls can overlap and you have infinitely many of them. Long answer:
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C|net: Just how random is random?

Friday, March 9th, 2007

C|Net accuses Apple of favoring iTunes songs over CD-ripped songs in iTunes random playlists. Unfortunately they don’t have the statistical chops to prove anything or do any real analysis:

It’s obviously difficult to tell whether back-room marketing deals or just dumb luck were responsible for the results we saw, but it appears that we can safely lend credence to the suspicions of myriad iPod users around the world. When it comes to choosing songs, ‘random’ clearly is relative.

Actually folks, it’s totally possible to figure out whether your results are random luck or not. For one thing, try repeating the experiment. But what you really need are better statistics. In particular try calculating the chance your results would occur by pure randomness. You haven’t published the raw data, so I can’t do it for you; but this should be well within the reach of anyone whose taken a couple of undergraduate courses in statistics. In fact, it would make a very nice final project for a statistics course. I don’t think it quite rises to the level of an undergraduate thesis though.
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