Recent content by joeboo

Careful here; the map between topologies is a correspondence between elements of the topologies, or open sets. The bijection between the spaces is a correspondence between elements in the spaces, or points. They are not the same functions. However, your above argument is valid IF the...

I was suggesting you consider the scenario where you have a bijection of sets in addition to an isomorphism of the topologies, and see if this is equivalent to the spaces being homeomorphic.

I believe so, yes, but how would you show it? If you have a homeomorphism between two spaces, could you then construct (using said homeomorphism) a isomorphism their respective topologies? Also, what do you think of the alternative I suggested? It may help you understand the situation better.

Consider any two sets of different size. Now give them both the indiscrete, or trivial topology (only open sets are the empty set, and the entire set itself). How does this affect the argument? Suppose you also have an underlying isomorphism of sets (ie a bijection). How does this change...

Perhaps I didn't explain myself properly ... What you are describing is exactly what I suggest, except you don't go far enough. Consider the part of your quote I bolded. What is 'nicely" ? For you, it may involve dissecting the object into smaller parts whose areas you can calculate. For...

Learning of all kinds involves memorization in some form. The question is, what is one memorizing when they learn? Certainly, a man could memorize every mathematical theorem, and definition, and be able to interpret them. However, this does not make him a mathematician, in my opinion...

t(x) = \frac{1}{\sqrt{\vert x \vert}} Am I missing something?

First off, let's do x = 2t, and set f(t) = 4^t - 3^t - 1. This, in my opinion, just makes the whole thing easier to work with algebraically. It is not a necessary change, but does make the steps easier. Hints for proving there is a unique zero of f(t): f(t) is clearly continuous, therefore...

When insecure people need to bolster their (non)beliefs with logic and reasonning that is provided by others because they aren't capable of doing the work themselves.

While this works perfectly for an illustrative example of why any definition of division by zero is "undefined", I think it is misleading. The actual reason why division by zero is undefined is because undefined is to be taken literally as meaning "It is not defined". This is because the...

Haha! Perfect, snoble, thank you. Now that I see it, I can't believe what I was thinking. I think at some point, I was equating the open ball with the closed ball in the uniform tolology. As in, it included the values for which the supremum was reached. Obviously, that's not the...

Haha. The letter "written" by Andrew Wiles is classic: Yep. That sounds like a letter an intelligent person would write: "And let me tell you" ... come on. Further, I can't tell if the second paragraph is sarcasm or not. The article also states: Note the bolded text ( by me )...

Let \rho be the uniform metric on \mathbb{R}^\omega For reference, for two points: x = (x_i) and y = (y_i) in \mathbb{R}^\omega \rho(x,y) = \sup_i\{ \min\{|x_i - y_i|, 1\}\} Now, define: U(x,\epsilon) = \prod_i{(x_i - \epsilon, x_i + \epsilon)} \subset \mathbb{R}^\omega I need to...

what is the \sqrt{mouse} ? or how about: \int{+d-} ?

( This is from an exercise in Munkres' Topology ) Let X_\alpha be an indexed collection of spaces, and A_\alpha \subset X_\alpha be a collection of subsets. Under the product topology, show that, as a subset of X = \prod_\alpha{X_\alpha} \overline{\prod_\alpha{A_\alpha}} =...