Dense
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From: William Elliot on 12 Apr 2010 08:24 On Sun, 11 Apr 2010, porky_pig_jr(a)my-deja.com wrote: > On Apr 11, 9:18�am, The Qurqirish Dragon <qurqiri...(a)aol.com> wrote: > >> It appears you are confusing measure with the concept of being dense. > > Yeah. That's rather amusing, isn't it? From the topological > perspective both Q and R\Q are dense. Yet, say, on [0,1] the > (Lebesgue) measure of R is 0 and the measure of R\Q is 1, so from the > measure perspective, R is practically non-existent. > Surely you mean Q is practically non-existent. Anyway now that you've clarified that Q is barely dense and R\Q is almost dense we can determine that cofinite sets are mostly dense.
From: William Elliot on 12 Apr 2010 08:47 On Sun, 11 Apr 2010, Achava Nakhash, the Loving Snake wrote: > On Apr 11, 5:08�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >> Thus the quandary: �what is a real dense set? > > Bad grammar. You should have said, "What is a really dense set?" > Not so. set ~\~~~~\~~~ \ r \ d \ e \ e \ a \ n \ l \ s \ \ e set ~\~~~~~ \ d \ e /\ n / \ s \ r \ e \ e \ \ a \ l \ l \ y \ Though you may claim "real dense set" is bad punctuation (missing a comma), you would miss the pun. In addition, there's slang useage real great real good real bad for really great really good really bad and misc., etc's though of recent bad can also mean good, like real good.
From: Robert H. Lewis on 12 Apr 2010 08:52 > It's claimed Q is a dense subset of R. > > How can that be? It's got more holes in it > than it has plugs. Why isn't it called porous? > If X is a topological space, a subset A is called dense if every nonempty open set of X contains a point of A. This is true for Q in R with the usual topology. Essentially, you need to show that between any two irrationals there is a rational. That's intuitively obvious, but to prove it, you need a formal definition of R. Similarly, the set of irrationals is dense in R. For this, you need to show that between any two rationals there is an irrational. That's easy. Robert H. Lewis Fordham University
From: ksoileau on 19 Apr 2010 05:17
On Apr 11, 3:48 pm, "porky_pig...(a)my-deja.com" <porky_pig...(a)my- deja.com> wrote: > On Apr 11, 9:18 am, The Qurqirish Dragon <qurqiri...(a)aol.com> wrote: > > > It appears you are confusing measure with the concept of being dense. > > Yeah. That's rather amusing, isn't it? From the topological > perspective both Q and R\Q are dense. Yet, say, on [0,1] the > (Lebesgue) measure of R is 0 and the measure of R\Q is 1, so from the > measure perspective, R is practically non-existent. Nope, you have it reversed. You should have said: Yet, say, on [0,1] the (Lebesgue) measure of Q is 0 and the measure of R\Q is 1, so from the measure perspective, Q is practically non-existent. |