Thread: box, product, uniform topologies and convergence

I am having a great deal of difficulty visualizing and grasping the concepts of box, product, and uniform topologies. Is there a clear, concise-ish way to explain it?

Also, I sort of understand the concepts of converging sequences under these topologies, but am lost. How does one determine whether or not a sequence converges with respect to different topologies?

Thanks.

Originally Posted by iamthemanyes

I am having a great deal of difficulty visualizing and grasping the concepts of box, product, and uniform topologies. Is there a clear, concise-ish way to explain it?

Also, I sort of understand the concepts of converging sequences under these topologies, but am lost. How does one determine whether or not a sequence converges with respect to different topologies?

Thanks.

Unfortunately we can't teach you a whole subject here. There is multiple ways of thinking about these spaces. You can just list their bases (subbases), you can think of the product topology as the induced topology from the set of maps etc. Can you ask a little more specific of a question?

For example, the sequence
(1, 1, 1, 1, ...)
(0, 2, 2, 2, ...)
(0, 0, 3, 3, ...)
...
converges in the product topology, but NOT in the uniform topology. Can anyone explain why this is?

Originally Posted by iamthemanyes

For example, the sequence
(1, 1, 1, 1, ...)
(0, 2, 2, 2, ...)
(0, 0, 3, 3, ...)
...
converges in the product topology, but NOT in the uniform topology. Can anyone explain why this is?

It's essentially the difference between pointwise and uniform convergence.

Convergence in the product topology is equivalent to coordinatewise convergence. The above sequence converges coordinatewise to (0,0,0,0,...). But for it to converge uniformly, you would have to show that, after a certain point in the sequence, all the coordinates of each point were close to 0, and clearly that is not going to happen.