# Recent content by gonzo

1. ### Continuous functions

the continuity is what says that the inverse image of any closed set is closed, and dalle wrote in his last post. My comment on the union of two closed sets was responding to that.
2. ### Continuous functions

Okay, I think I'm making progress now. If G_E \in (E,\mathbf{E}) is open, then we have G_E = G \cap E for some open set G in X. This means that G_E^c=G^c \cap E where G_E^c is the complement of G_E in E and thus an arbitrary closed set in E. Since E is closed and the intersection of two...
3. ### Continuous functions

Thanks, dalle. That seemed to be helpful, but I ran into a problem. 1. I have as a theorem I can use directly I thought I could prove 2. easily, but the problem was it didn't seem to depend on E or F being closed. Then I seem to be making some leap again to the final proof that doesn't depend...
4. ### Continuous functions

Okay, we can forget my example. It was a bad example. But still don't see how to start on a positive proof. I was able to do it for E and F open, but I seem to run into problems for them both closed.
5. ### Continuous functions

Obviously the continuity is the limitation on the topologies here and the problem must turn on that point. But I don't see how to work out a proof from that (even if I could maybe see how it applies to my specific bad example).
6. ### Continuous functions

That doesn't help me.
7. ### Continuous functions

I'm having trouble with the third part of a three part problem (part of the problem is that I don't even see how what I'm trying to prove can be true). The problem is: Let X and Y be topological spaces with X=E u F. We have two functions: f: from E to Y, and g: from F to Y, with f=g on the...
8. ### Int(A) + ext(A) not dense

Nevermind, I got it. I forgot about the possibility that int(A) and ext(A) could both be the empty set, and thus the closure would also be the empty set (which I guess meets my criteria anyway of int(A) = ext(A)).
9. ### Int(A) + ext(A) not dense

Sorry, bad phrasing. I figured it should have been obvious what I meant from context. I believe my book calls them "interior points", which are points that are contained in some open set that is completely contained in the set in question.
10. ### Int(A) + ext(A) not dense

int(A) = interior of A ext(A) = exterior of A, or the interior of the complement of A My thoughts are thus: int(A) and ext(A) are both open sets, so their union is an open set, and if we let B = union of int(A) and ext(A) then B = int(B). So the only way the closer is not equal to the entire...
11. ### Int(A) + ext(A) not dense

Can someone help me find an example of how the union of int(A) and ext(A) doesn't have to be dense in some space X? Thanks.
12. ### An integral for thought

Is this complex or real integration? And what are your limits of integration? And it would probably help if you wrote out the entire integral.
13. ### Rational points on a circle

I already posted the basis of my proof, the rest is trivial. Pick two rational points (points with rational coordinates) on the unit circle: e^{i \theta_1} and e^{i \theta_2} Then it is really easy to show that the distance between the points with double the angles e^{2i \theta_1} and e^{2i...
14. ### Rational points on a circle

Thanks jing. But like I said, I can't take credit for it. Wish I could since it was really obvious and simple when I saw it. But I can't.
15. ### Rational points on a circle

The proof I have for the unit circle (and it's not my proof, I can't take credit for it) doesn't directly extend to circles with non-perfect square radii. However, there is another proof I've seen that might. Why do you ask?