Name: mod07lec43 - Proof of Tychonoff's theorem - Part 1 | 影片 | 乾淨世界
Uploaded: 2023-12-02T17:19:17.271Z
Duration: 28 min 47 s
Description: We finally give a detailed proof of Tychonoff's theorem, using Zorn's Lemma in a fundamental way. As a Corollary, one obtains the fact that (1) a finite Cartesi... | 影片 | 乾淨世界

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https://video5-us-east.cloudokyo.cloud/video/v6/2c/93/cf/2c93cffb-8017-4434-a464-68fc0bcbef55/master.m3u8

We finally give a detailed proof of Tychonoff's theorem, using Zorn's Lemma in a fundamental way. As a Corollary, one obtains the fact that (1) a finite Cartesian product of compact spaces is compact, and (2) any closed and bounded subset of R^n is compact, since it is a closed subset of a product of compact intervals in R (which is then compact by (1). We note that both these points can be proved without the full force of Tychonoff's theorem, but here we give them as a corollary.