Both questions are rebutted by the same observation: a topological space is not a set. A topological space is a set and a specified topology.
A line in R[sup]3[/sup] and a plane in R[sup]3[/sup] with the subspace topologies induced from the metric topology on R[sup]3[/sup] are not homeomorphic. That wasn’t the statement, however. The statement was just that the line and the plane aren’t homeomorphic.
Similarly, considering a cylinder as an appropriate étale bundle over the circle (as opposed to the “standard” product topology as a circle crossed with a line) makes them homeomorphic. It’s very similar to the Riemann surface of the logarithmic function over the Riemann sphere: an infinite cyclic cover.
Oh, please. Topology doesn’t exist in a vacuum, and for that reason there is such a thing as a standard topology on those two spaces. I repeat: continuity on the reals can be and is defined long before the word “topology” ever has to be uttered. In fact the entire notion of a topology is an abstraction which was inspired by the notion of continuity of real-valued functions. So it’s entirely possible for a non-topologist to ask “are the line and the plane homeomorphic”, and when a non-topologist does ask such a question we don’t have to bend over backwards to guess what topology they mean. Abstract topologies are a good and wonderful thing, but let’s not forget what they’re an abstraction of.
Unfortunately, that project has been completed, with null results. If the Universe has a nontrivial topology, the elemental cell is larger than the observable universe, and hence no repetitions are visible to us. This was actually suspected from the start, but a non-null result would be interesting enough and the effort required was small enough (since we already had the data anyway) that it was considered worth the investment of time despite the low chance of payoff.
Cabbage, you forgot one condition in your definition of a metric, that the distance be zero only if the two points are identical. Otherwise, the trivial function where f(x,y) = 0 for all x and y would be a metric, which it is not.
For the OP’s questions, all I can add is that you really, really never want to have to solve a nonlinear PDE. They almost always require a computer, giving you only a numerical solution, and even there, you generally have to be really careful in coding, based on the details of the particular PDE you have, to make sure that your program is stable, or you’ll end up with a solution in which the computer is very confident, but which is absolutely wrong.
Oh, I was well aware that these things were going to be nigh on impossible for me to understand based on my lack of sufficient math to grasp any explanation that’s not a gross simplification.
The friend who first told me of PDEs and topology (part of an elaborate scheme to meet girls, if you were curious. It failed) said, “since people are actually answering you, how about you see if they can take a whack at explaining number theory?” So, how about it everyone? Number theory?
The same friend (who is a math major) said something to the same effect. Something about “I’ve done linear PDEs and they are God-awful. I can only imagine that non-linear ones would be about the most horrible thing ever”
Sincere apologies for the wording of the title. Maybe I’ll learn to stop posting at 1:something in the AM
It failed? Huh, who could’ve seen that one coming?
Anyway, number theory is (at bottom) the study of numbers–namely, the complex numbers and the notable subsets thereof–with the goal of discovering facts that are slightly more sophisticated than 1 + 1 = 2. The Goldbach conjecture is the most famous number theoretic problem, and understanding it doesn’t require anything above grade-school math.
Complex numbers come into number theory, but I tend to see them used more as a tool than an object of study themselves. It spins out of the study of the ring Z of integers, specifically its ideal theory.
No, we don’t have to bend over backwards, but it behooves us to be explicit in our language and terminology to avoid causing complications with their later studies.
I can see where my students’ multivariable calculus text shamelessly identifies R[sup]3[/sup] and the space of tangent vectors at a given point. This is indeed canonically possible, but the generalization (to manifolds) requires keeping the concepts seperate. In case they ever go into physics, I make a point of keeping tangent vectors and points distinct, even though there is a canonical identification.
Similarly, “line” or “plane” are not properly specified as topological spaces, even though there is a “natural” topology to put on them as far as newcomers to the field are concerned. Still, it should be made very explicit that we are picking a topology out of many that could be placed on the spaces.