I'm sorry, I didn't realize my question was ambiguous.
But if there are no significant differences between metric and topological spaces, then your understanding is correct. Thank you for your answers. I'll try to be more specific in the future.
Well, I'm out of my league here. I'm only in my first undergrad year and I haven't taken any topology yet (although I have Munkres' book and intend to put it to good use in the summer), so I'll have to study some before a proof or counterexample along those lines will make sense to me.
Does that equation have any real solutions? Or even complex ones?
We know that for all x that |cos x| <= 1, and therefore |cos^n x| <= 1. So based on that we get
cos x + cos^2 x + cos^4 x <= 1 + 1 + 1 = 3 < 4
so we must conclude that the original equation has no solutions.
By the way, I first tried arguing that since f sends compact sets into compact sets, f also sends closed sets into closed ones. This doesn't hold because it says nothing about a closed set which isn't totally bounded and all bets are off on that one.
Unless are no closed sets which aren't...
Hmm... that's strange. I thought I managed to prove this earlier. Here's how:
Let (M,d) and (N,r) be metric spaces, and f:M -> N a one-to-one and onto function. Assume that for every subset K of M holds
K compact in M <=> f(K) compact in N
Let's show that f is continuous.
I'm taking a course in analysis and I was thinking about the relation between compact sets and homeomorphism. We know that if f is an onto and one-to-one homeomorphism then it follows that for every subset K:
K is compact in M <=> f(K) is compact in N
Now, does this go the...
There's another fun variation on this theme where you line up all the numbers from one to nine in threes and are supposed to make them add up to six by adding only plus, minus, division, multiplication, root and power signs (whole powers and roots, no logs!). You can also use ( and ) (forgot...
I'm sorry, I'm from Iceland and math is tought in icelandic, so I don't know all the english names for functions.
I'll look over what NateG posted tomorrow (kind of late here at GMT), since I've run into trouble deriving it myself the least I can do is learn how to do it properly. Thanks guys.
sin(-x) = -sin(x)
this can be seen as this example:
sin(i^2x) = i^2sin(x)
Does this then apply?
sin(ix) = isin(x) ?
I'm trying to derive a formula for the length of a simple parabola. Unfortunetly in my calculations I end up with Arcsin(i2x) along the way and it...
Cryptography is a complex field, you can't really sum up the methods of it in one equation (At least I can't), but there's an excellent book available by Simon Singh that's called "The Code Book" that tells you how cryptography was invented, how it was developed, how the modern cryptography...
I think one of my books mentioned a way of solving third level equations (ax^3 + bx^2 + cx + d) and fourth level equations (Same as before, add nx^4) much the same way as you do with second level equations ((-B +- Sqrt(B^2 - 4AC))/2A). I have two questions, do you guys know the formulas for...