How You Should Think About Infinite Cartesian Products
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- Опубликовано: 8 фев 2025
- I regularly teach a topology class online, and often it's the first time my students encounter infinite Cartesian products of sets. This video aims to help you understand what elements of such products look like.
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Amazing visualization. When I first studied topology, I thought of R^w (or R^N) as the set of all real-valued sequences and R^w1 (or R^R) as the set of all real-valued functions. Later I learned about how this concept is called an Exponential in a Cartesian Category. Studying category theory really helped me go back and re-learn abstract algebra, topology, etc. through a new lens.
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very good demonstration 👍🏻 really looking forward to more videos from you.
Thank you!
Wonderful. Thanks
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Pretty good video. i never thought of it like this before
Thank you!
I should have seen this video when I started studying topology on infinite cartesian products, it would have made much more sense to me. Great video, as always :)
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I just finished learning about co-vectors which are the set of linear functions from R^n and has size R^n. I wonder what the size of other restricted sets of functions like polynomial functions or continuous functions ect.
Hi Tom, that’s such cool stuff! Each of those sets has the same cardinality as R.
I wonder how do you animate these things?
Using Manim.
aren’t you assuming CH here? or am i misinterpreting the notation
I don’t think so. CH says that the cardinality of omega_1 equals the cardinality of the reals. I believe I said the cardinality of omega_1 is at most the cardinality of the reals, but I did not say they’re equal.
@ right, thanks!