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# 265

Oral Exams - July 7, 2010
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I heard this joke at a conference one time :D

Looks like the poor guy is gonna flunk.

This reminds me of my own candidacy exam back in the late 80s. I got the vibes that I had done really well in algebra and adequately in complex analysis, so when the first real analysis question was "Name an uncountable set of Lebesgue measure zero" I couldn't resist and replied: "Ehh... can I do it in the plane?" At that point the panel of professors burst out laughing (while I was also crying "Cantor set!"), so I knew that I had them on my side (which is the case more often than not).

Good luck to all the readers still preparing for their orals!

I wish you had revealed the guy's expression in the first panel when the examiner says he's screwing up. I love the cute sad faces you draw.

Don't remind me, I have another one tomorrow... I flunked one of my four in, well, just about the same manner shown here. So... get out of my head :)

To be clear, I could've told you the right answer before and afterward, but I always freeze up in oral exams. :P

Why, trivial, of course!

The best oral exams end up as a mutual discussion of interesting ideas...the problem is getting to that point.

for me the term "oral exam" has always sounded dirty, I really need to get my mind out of the gutter

Just curious: Under what topology is the set of real numbers compact?

The finite complement topology should work.

Which is also the Zariski topology (under which also R^n is compact).

also the trivial one. or the one obtained by bijecting them with [0,1] homeomorphically (define the topology to make the bijection homeomorphic).

I always check both the trivial and discrete topologies on specific or arbitrary sets for examples and counter-examples.

Sorry I don't get the joke (BSc only :) Can someone explain?

that's not a math BSc, right?

In full generality, "compact" requires some background. (In a topological space S, a subset Q is compact is every cover of Q by open sets in S contains a finite subcover.) However, on "well behaved" sets one can use the shortcut "compact" = "closed and bounded". "Closed" here means "contains its limit points". On well behaved sets, "bounded" means sits inside a ball of some specified radius centered at the origin.

If R under the usual topology is the "big" set and R is the selected subset, then R is closed, open, both, neither, whatever. However, R is not bounded in R (under the usual topology). So R is not compact.

This suggests that under an unusual topology R might be bounded and therefore compact.

Nevertheless, *much* better answers would have been "the unit interval in R under the usual topology", the "unit disk in C under the usual topology". Or even, the set {x} (i.e. one element) under pretty much any topology you like.

Boundedness only makes sense in a metric space, and the "compact" = "closed and bounded" works only for subsets of R^n (Heine-Borel). But it is true of course that compact sets in metric spaces are bounded, and also that they are closed (metric spaces are Hausdorff).

In general, a metric space is compact, iff it is complete and totally bounded.

So essentially the joke is that the student gave a pretty bad answer to an easy question and the examiners are looking for edge cases where he might have been right. There isn't some other wordplay that I'm missing, is there?

lol I messed up the multi-variate Gaussian pretty bad in my orals last month :P

the explanation regards the Heine–Borel Theorem right???
Wikipedia knows it all
http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem

For my S.T.B. (Theology) degree, we had the option of three hours of written Comprehensives, or a one hour oral Comp. The faculty advised strongly that students do the written one: most students seemed to do more poorly on the orals.

Of course, you don't get to chose questions in orals, and have less time to think things through, and once the examiners see you know what you are talking about, they can quickly move on. Much easier to do the written test, even if your hand will be ready to cramp after the first hour or two...

Poor girl. I believed she flunk ;)

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