May 13th, 2021

On the point-set topology pertaining to Tychonoff’s theorem

Originally published at 狗和留美者不得入内. You can comment here or there.

We defined a base of a topological space in Definition 4 of [1]. In short, a base via unions generates the topological space. Since the intersection of finitely many open sets is also an open set (this follows directly from one of the neighborhood axioms), one naturally also defines a set that generates the topological space per arbitrary unions of finite intersections. We thus have

Definition 1 A subbase of a topological space is a collection of its open sets, [公式] , such that for every open set [公式] , there exists a finite subset of [公式] , [公式] such that [公式] .

Proposition 1 A base of a topological space is also a subbase of it.

Proof: Trivial. [公式]

Since Tychonoff’s theorem involves product spaces, we shall also define them.

Definition 2 Let [公式] and [公式] be topological spaces. [公式] is continuous iff for every sequence [公式] in [公式] that converges to some [公式] , the sequence [公式] also converges to [公式] . Equivalently, if [公式] is open in [公式] , then [公式] is open in [公式] .

We leave the proof of the equivalence in Definition 2 to the reader.

Definition 3 Let [公式] be a collection of topological spaces indexed by [公式] for [公式] where [公式] is an arbitrary set. We take [公式] to be the Cartesian product of all the topological spaces in the aforementioned collection. We let [公式] be the canonical projection to the component corresponding to index [公式] , which we require to be continuous for all [公式] .

Proposition 2 Every topology on [公式] must contain, for all [公式] open in [公式] for all [公式], [公式] .

Proof: This must hold if the canonical projections are to be continuous. [公式]

Definition 4 We define the product topology or Tychonoff topology on [公式] to be the coarsest topology on it such that canonical projections are continuous. This is the topology generated by [公式] via unions of finite intersections. In other words, this is a subbase of the product topology. One can also say that the open sets on [公式] are the subsets of it such that all but a finite number of the canonical projections [公式] across all [公式] , when applied to the subset in question, yield [公式] (the codomain of [公式] ), and that those which do not yield the codomain, are all open (in the topological space defined on the codomain).

Tychonoff’s theorem (Theorem 1) The topological space on the Cartesian product of any collection of compact topological spaces with the product topology is compact.

Proof: Definition 4 tells us that [公式] is a subbase for the product topology. Take any cover [公式] of the product space by elements of this subbase. We assume [公式] does not contain [公式] , in which case Tychonoff’s theorem holds trivially. With the exception of [公式] , every element of this subbase has exactly one [公式] for which the projection onto the [公式] component is not the [公式] . This induces a map from the subbase to [公式] . Thus, we can partition [公式] into the disjoint union [公式] , where [公式] is the preimage of [公式] of the aforementioned map. Suppose that no [公式] covers the space. Then for each [公式] we pick an [公式] not such that [公式] , which gives a point in [公式] not in any of element of [公式] , which would imply that [公式] is not a cover, a contradiction.

We note that [公式] covers the space for some [公式] and [公式] , where every [公式] is equal to [公式] for some [公式] open in [公式] . We see that[公式] is an open cover of [公式] , which, being compact, has a finite subcover, which we denote as [公式] . Then [公式] is a finite subcover of [公式] , which covers the product space. This shows that if we can prove that a topological space such that every cover from a subbase is compact, we have proven Tychonoff’s theorem. The aforementioned proposition is called the Alexander subbase theorem, named after American mathematician James Waddell Alexander II, which we shall prove next. [公式]

Alexander’s subbase theorem (Theorem 2) If every open cover from a subbase of a topological space has a finite subcover, then the topological space is compact.

Proof: We denote the cover by elements of the subbase as [公式] . We also remark that any open cover is a collection of sets and that there is a partial order on open covers via collection inclusion. Assume the hypothesis of the theorem, and then assume by contradiction, there there are open covers which do not have finite subcovers. Take all the collection of all such open covers. Let [公式] for [公式] be any collection of covers without finite subcovers and [公式] a total order such that whenever [公式] for [公式] , [公式] . Let [公式] . If [公式] had a finite subcover by sets [公式] , then we can find [公式] such that [公式] , in which case the maximum of [公式] per the total order defined on [公式] , which we denote as [公式] , is such that [公式] has a finite subcover, a contradiction. This means that every chain in open covers without finite subcovers is bounded above by some open cover without a finite subcover. This enables us to apply Zorn’s lemma (proven in [2]) to derive some open cover with finite subcover that is not a proper subset of any other such open cover, which we call [公式] . By our hypothesis on the subbase, [公式] is not a cover. Thus, we can take an [公式] in our topological space not contained by [公式] . Any [公式] containing [公式] is not in [公式] . For any [公式] where [公式] , [公式] must contain a finite subcover, which we denote as [公式] . [公式] , which also means that Let [公式] . By the definition of subbase, there exists [公式] that such [公式] and [公式] . Let [公式] be a finite subcover for the cover [公式] for all [公式] . Then, [公式] would be a finite subcover of [公式] , a contradiction. This completes our proof. [公式]

I shall now explain a bit of the intuition behind the proof of Tychonoff’s theorem using the Alexander subbase theorem. I believe it should be more or less immediately for anyone who has studied boolean algebra deeply.

A difficulty some students might have with point set topology is the existence of not only countable unions or products but also of uncountable ones, as in the general Tychonoff’s theorem. We note how here de Morgan’s laws still apply. A element’s not being in an countable union of sets means it is not in any of the uncountably many unioned sets. One, after putting the NOT or set complement operator inside into every set unioned also switches the union to intersection. In the proof of Tychonoff’s theorem, this immediately tells us that the proposition of not being a cover evaluates to true iff a conjunction of nots of set inclusion evaluates to true, which can only be violated if no [公式] covers the space. This is basically saying that the “true for all” operator applied to a collection of boolean values evaluates to true iff all of the elements of the collection are true. From this, we can see that in general, if we want to prove something given the hypothesis of membership in a union of sets, it suffices to prove anything that implies non-membership in all the unioned sets of the element in question.

Inclusion of an element in a Cartesian product of sets a logical conjunction operator of the inclusion of the projections into their corresponding sets. [公式] lets us isolate to a component of the product space to give us a covered guarantee based only on the value of one component. If the value of the that component does not lie in the guaranteed range of values, it is covered iff the value of some other component lies in the guaranteed range of values for that other component. If [公式] for some [公式] , that guarantee obviously always exists. If that doesn’t hold for any [公式] , then there exists an uncovered point.

References

More on the unrealistic expectations of Chinese-Americans

Originally published at 狗和留美者不得入内. You can comment here or there.

This is in response to http://disq.us/p/2h00fdj.


我那朋友好像跟我说了那个人。

This reminds me that Steve Hsu thought in 2019 that the Chinese in computer science suck, including the ones in America. He was like,

Name a single programming language created by a Chinese!

I actually looked a bit into this and on Chinese Wikipedia, I found information about this programming language developed and used in the 60s in China, which was a variant of ALGOL. I also found online some Chinese papers published in the 1970s involving indigenously developed programming languages for indigenous operating systems. The guy who created the first operating systems in China was a son of a mathematician who was SS Chern’s master’s advisor, who did some years of exchange at a computer company in Britain in the early or mid 60s. Interestingly, the world’s first operating system was created in Britain, not in America, in 1961. To be fair, operating systems and programming languages and compilers are not that hard. Programming language is mostly a tool. I talked about this with a friend of mine in China who’s published pure math papers. He thinks that once the programming languages satisfy the needs, there isn’t much point in developing more of them. Knowing a bit of type theory myself and having written a toy compiler for a class with a partner in undergrad, I do think programming language theory is very interesting in its own right. I don’t think it’s really all that useful, and I can totally see why there was hardly anyone investigating it seriously in China between 60s and 80s, when the country was poor and resources were limited. There was pretty much no theoretical computer science in China before Andrew Yao in 2000s, maybe because theoretical computer science is for the most part actually a rather useless field, though computability and complexity theory are interesting, though also somewhat fringe even by pure math standards. Maybe that’s partly why Berkeley math department denied Stephen Cook tenure.

I’m also pretty critical of Fields Medalist Shing-Tung Yau’s talking all the time about how China hasn’t produced many great mathematicians and criticizing the Chinese education and academic system that he’s never been genuinely directly involved in. Interestingly, he told a Westerner interviewing him that his father, after fleeing in 1949, actually believed that the KMT would successfully counterattack within a couple years, after which they would be able to return to mainland, which indicates that his father was delusional.

Yau also aggressively lobbies the Chinese government to invest $200 billion to build a supercollider for high energy physics. He is utterly ridiculous. More evidence that overseas Chinese, including the scientists and engineers, are at the core opposed to the interests of the real Chinese in China.

By the way Princeton and Berkeley professors Wu-Chung Hsiang and Wu-Yi Hsiang, brothers and kids of KMT officials who fled to Taiwan, also pretty much hate Yau. Wu-Yi exposed Yau’s ridiculous behaviors in a talk at Beijing University, which Yau fell out with in the 2000s.

I wrote a piece criticizing Yau, referring to Yau’s utter political stupidity as indicated by his words in the linked video. His reputation in China is already pretty much ruined, and he’s making it worse by saying something as stupid as that in a video publicized online. In that piece I wrote, which one can find on my blog, I bluntly noted that militarily forcing out the KMT reactionaries to Taiwan is much more significant than the production of a few more great Chinese mathematicians in China, which should be obvious to anyone with the slightest bit of common sense. Yes, I love theoretical science, but I don’t treat it as a “sacred” enough to lose my common sense, as some nutters in academia do.

It’s also no surprise and entirely expected that the Chinese who stayed in America after 49 did much more at the forefront of science and technology (excepting military technology) than the Chinese in China. It obviously doesn’t mean they are more talented or inherently better than the ones with American graduate degrees who returned. That generation and their kids in their long term lost their social status by immigrating. China won’t really accept them as their own either. Unlike the ones that returned, they didn’t accept that the reality by being from a then much weaker country and ethnicity, they in some sense had “debts to pay”. For their own careers or economic living standards, they implicitly sacrificed the prospects and status of the next generation(s). In this regard, it’s fair to say that they are lesser than the ones who returned to China in terms of character and overall ability. Science and technology ability and achievement is not everything. A analogy I like to make is that a kid from a family of limited means shouldn’t be doing expensive sports to come across as “cool” after seeing a cool rich kid do that.

This is of course not unique to Chinese. The baby boomer generation in America, who had it so easy in comparison, also pretty much screwed over the prospects of the millennials. They are much to blame for America’s decline. At the end, the collective society and its ethos matters much more than a few individuals at the top.