Name: General Topology
ISBN: 9780923891558

What Hacker News says

graycatJun 2024

One relatively simple but informal description of the problem is that the statement is self-referencing.

Related is the Bertrand Russell paradox:

X = {z | z is not an element of z}

So, X is an element of X if and only if it is not -- a contradiction.

The usual solution was to have two steps:

(1) Over here, say, on the left, we have everything we want to regard as elements.

(2) On the right, we have everything we want to regard as sets. The sets consist only of elements.

That's a good fix.

For more, there is the elegant, long popular, P. Halmos Naive Set Theory, now available in PDF, LaTeX, etc.

For more there is the appendix to J. Kelley, General Topology.

For more there is P. Suppes, Axiomatic Set Theory.

For more, sure you want more????

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yomritoyjApr 2018

Always within arm's reach at both home and work:

Rudin, Real and Complex Analysis

Halmos, Finite Dimensional Vector Spaces

Billingsley, Probability and Measure

Kelley, General Topology

Lang, Undergraduate Algebra

Simmons, Introduction to Topology and Modern Analysis

Parthasarathy, Probability Measures on Metric Spaces

Whinston, Green and Mas-Colell, Microeconomic Theory

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graycatApr 2018

A distribution is a real valued function of a real variable. The domain of the function is the whole real line.

Note: Below, borrowing from D. Knuth's TeX, we use the underscore character '_' to denote the start of a subscript.

Details: For real valued random variable X, probability measure P, and the set of real numbers R, the cumulative distribution of X is the function F_X: R --> R where, for x in R, F_X(x) = P(X <= x).

If F_X is differentiable, then the probability density distribution of X is the real valued function of a real variable f_X: R --> R where, for all x in R, f_X(x) = d/dx F_X(x) where d/dx is the calculus derivative.

For the connections with sigma algebras, that is more advanced than most engineers care about, but here are some of the details:

For real numbers a and b with a < b, there is the open interval

(a,b) = {x|a < x < b}

A topology on R is a collection of subsets regarded as open and that satisfy the axioms for a topology -- the sets in a topology are closed under finite intersections and arbitrary unions and both R and the empty set are open. The usual topology on R is the smallest (a short argument shows that this "smallest" is well defined) topology that has each open interval an element of the topology -- right, the topology regards the open intervals as open.

The usual reason to discuss a topology is to have a means of defining continuous functions, a means more general than from the usual "for each epsilon greater than zero, there exists a delta greater than zero such that ..." or in terms of limits of sequences. Indeed, there are advanced situations where we can use topologies to define continuous functions where epsilon and delta and where converging sequences don't work. If curious, look up Moore-Smith convergence, nets, and filters or just Kelley, General Topology.

Well, a sigma algebra is like a topology, that is, is a collection of subsets: A sigma algebra is closed under countable unions and relative complements. Right, we avoid uncountable unions because otherwise we will get stuck in a big mud hole. It is an early exercise that there are no countably infinite sigma algebras.

The reason for sigma algebras is to permit defining a measurable function, that is, one where we can apply the Lebesgue integration theory. The integral of calculus is due to B. Riemann and is the Riemann integral. W. Rudin, Principles of Mathematical Analysis shows that for a continuous real valued function with domain a compact set (closed and bounded, where closed is the complement of an open set) has a Riemann integral. Well in this case, the Lebesgue integral gives the same numerical answer -- same thing. The advantage of the Lebesgue approach is that the function can be even bizarre and its domain can be much more general. Indeed, in probability theory, expectation is just the Lebesgue integral. In simple terms, Riemann partitioned on the X axis, and Lebesgue partitioned on the Y axis.

Well, given the usual topology on R, we can ask for the smallest sigma algebra on R that has the topology as a subset. That sigma algebra is the Borel sets of R. Uh, Lebesgue was a student of E. Borel. In Rudin will find the Heine-Borel theorem.

So, in probability theory, we have a sample space. Each point in the sample space is a trial, i.e., essentially a real world experimental trial (note: really our attitude is that in all the universe we see only one such trial -- if this seems far out, then blame the Russians, e.g., A. Kolmogorov, E. Dynkin, etc.!). Well, an event is a subset of the sample space, that is, a set of trials. So, flip a coin. Let H be the event, the set of all trials where, that the coin comes up heads.

Well, to apply Lebesgue's theory of integration, we want the set of all events to be a sigma algebra.

Then a probability measure is a measure in the sense of Lebesgue's measure theory, that is, a real value function, in the case of probability taking values in [0,1], and with domain the sigma algebra of events. So, for the event H, we can ask for the probability of H, that is, P(H), which is a number in [0,1]. For a fair coin tossed by an honest member of the FBI we have P(H) = 1/2.

Then a real valued random variable X is just a real valued function with domain the sample space and also measurable: This part about being measurable is that for each Borel set A, a subset of R, the set of all trials w so that X(w) is in A is an event, that is, an element of the sigma algebra on the set of trials. That is, the inverse image under X of the Borel sets are events, elements of the sigma algebra on the sample space.

So, with X measurable in this way, we have a near perfect shot at defining the expectation of X, E[X]. For this we have a little two step dance:

First we look at X^+ ('^' denotes a superscript) where X is >= 0. So, X^+ is the positive part of X. Similarly X^- is the negative part of X. So, X = X^+ + X^-. Uh, I'm working quickly from memory; maybe we want X^- to be -X where X < 0 and 0 otherwise. So both X^+ and X^- are >= 0 and we have X = X*+ - X^-. Either way.

Well, we can use Lebesgue's theory to integrate X^+ and X^-. Biggie stuff: The X need only be measurable, and that admits lots of really wildly bizarre functions. We've got great generality, and that's good to have in various limiting arguments. Uh, we like limiting arguments because that is our main way to approximate which our main way to being healthy, wealthy, and wise!

So the Lebesgue integral of X^+ we write as E[X^+]. Similarly for X^-. Now no way do we want to be subtracting one infinity from another since permitting that would trash the usual laws of arithmetic.

So, for our second step, in the case X^- >= 0, if at least one of E[X^+] and E[X^-] is finite, then we define E[X] = E[X^+] - E[X^-].

Now we've defined expectation ("average") of a real random variable X. Our definition is just the Lebesgue integral. For the Lebesgue integral, we wanted the sigma algebras.

On the real line we can consider the sigma algebra of Lebesgue measurable sets; that's larger than the Borel sets. Then we just ask, assume, assert, believe, ..., that our random variables are measurable with respect to the sigma algebra of Lebesgue sets and the sigma algebra of the events. Uh, right, Lebesgue measure on R assigns Lebesgue measure b - a to interval (a,b) and extends from there. Fine details are in various texts by Rudin, Royden, etc.

That's the beginnings of the role of sigma algebras in advanced approaches to probability, statistics, and stochastic processes. It turns out, the sigma algebra approach is for several parts of what we want in probability, much nicer, e.g., for defining independence and conditional expectation. E.g., if we want to know that some set of uncountably infinitely many random variables are independent, we can. Same for conditioning on uncountably infinitely many random variables, e.g., the past history of a stochastic process.

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graycatDec 2024

Another view of topology is in

John L.\ Kelley, {\it General Topology,\/} D.\ Van Nostrand, Princeton, 1955.\ \

In the set R of the real numbers and x, y in R with x < y,

(x,y) = { z | x < z < y }

is open and, with x <= y,

[x,y] = { z | x <= z <= y }

is closed.

A subset of R that is both closed and bounded is compact, a powerful property, e.g., in Riemann integration.

And so forth but in topological spaces much more general than the real line and open and closed intervals. Apparently hence the "General" in the book.

As a math major senior in college, read Kelley and gave lectures to a prof. But now there are some other definitions of topology.

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graycatJun 2022

Yes, I went to grad school in math. The first test was a pop quiz in real analysis and was about the little axiomatic set theory the prof had started the course with.

When the prof handed back the papers, he'd given me a zero. At the end of class, he asked that I stay. We went over my test paper: I had used little omega as the first infinite ordinal, essentially the same as the set of natural numbers. In an NSF course in axiomatic set theory the previous summer, that notation was standard. I'd thought of my proof, that is, my solution on the test, only at the last moment so just used little omega without defining it. When I gave the prof the definition, he saw that my solution was correct and, indeed, one step shorter than his. So, I got credit on the test.

Then I asked the prof why he had wanted to see me, and he smiled and said that he no longer did.

It was a class of 20+ students, and I doubt that all the other students got a good proof. So, why was the prof picking on me?????

He was regarded as a bright prof. In later years, he did have some fame. So much for such a bright math prof. I smelled I was being dumped on. I didn't know that prof at all.

His lectures were not very clear. The book he had selected was just in typing which is awful for the math of real analysis -- he could have used, say, Royden. And he'd just tried to dump on me for no good reason. Also, I was correct and he hadn't known that -- my meaning for little omega is and was standard.

At times I'd been dumped on in grades 1-12. I did well on standardized tests in math and physics, and that saved me. I was well out of patience being dumped on so walked out of the course and never saw that prof again.

In high school plane geometry the teacher believed that I refused to do any homework. Well, I didn't bother with her homework assignments -- they were too easy. Instead in the back of the book there were lots of more difficult exercises, and I made sure I worked 100% of those, never missed even one. One of those took me the weekend, and when I mentioned it in class on Monday, my first and last class participation, 20 minutes later the teacher was close to screaming exhorting the class to "think". Not wanting to be accused of ruining the class, as I started to give the steps to a solution, the teacher cut me off and screamed "You knew how to do it all along." Guilty as charged! Gee, she was not interested in getting the solution from me!

I've come to suspect: Students with some flair for originality and creativity can look different, not the same as the image of a good student -- nose to the grindstone, ear to the ground, shoulder to the wheel, and from that position dot all the i's and cross all the t's -- and less good than desired instead of better.

The other two courses they had me in were close to what I'd already studied carefully in ugrad school. One of these two was from Kelley, General Topology that I had lectured from to a prof.

I did well at my ugrad teaching, started violin, met my wife, and got a Ph.D. in applied math later at another university.

There, too, at times I did work comparable with that of the brilliant profs.

Net, I'm not so impressed by the super brains of so called brilliant profs.

Or I believe in the standard, "Everyone puts pants on one leg at a time."

More generally, my experience is, given the basic context of data and background, a lot of people can work through and find the immediate consequences.

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graycatApr 2021

The Kelley, General Topology pure math book acknowledges NSF and Office of Naval Research support in years 1950-1952. Same financial supporters for Luenberger, Optimization by Vector Space Methods -- fun with the Hahn-Banach theorem.

Good to hear that the NSF has supported some computing hardware -- my guess was that that was done mostly by the US Department of Energy. E.g., as I recall LINPACK was developed at Oak Ridge.

At one time I was teaching computer science at Georgetown U., and some colleagues wanted to work in speech recognition, asked for NSF support, and were told that the NSF did not support "software development" or some such.

Most of numerical analysis I saw was good definition, theorem, proof math. But recently I saw that the Formula 1 auto racing teams have been using CFD (computational fluid dynamics) for detailed design of the shape, wings, downforce, etc. of their cars. That's a lot of progress in fluid flow since I was working with the Navier-Stokes equations. So, apparently numerical methods for fluid flow have made a LOT of progress. As I recall, at times there was some such work at Courant Institute. I hope the NSF supported some of that progress.

For NSF support of "applying mathematics to biological problems" very good to hear, but my guess would be that that came under NSF support of biology instead of applied math.

Here is more what I had in mind: Daily the many US research-teaching hospitals take in sick people and make them well. There is good research, but the actual patient care is clinical, that is, serving people, and professional, e.g., with apprenticeship training, code of ethics, and liability and for professional practice. In the US I don't see anything similar for applied math or computer science, with or without NSF support.

Generally US research universities have a huge fraction of their annual operating budgets from grants from the US Federal Government, especially NSF and NIH. Roughly professors apply for grants, and about 60% goes to overhead for the university and, thus, also supports the English department, the string quartet series, the drama company, the alumni magazine, etc. Since the research universities don't much like math applications outside of academics, that NSF/NIH support is not much for such math applications.

E.g., university math departments have lots of talks by professors presenting their solutions without known non-academic problems but nearly no talks from non-academic people with problems looking for solutions. So, such academic math departments with NSF grants are not using that NSF funding to help non-academic people with problems find solutions.

E.g., in grad school in an applied math department, I studied a lot in optimization and stochastic processes, but I was the only one there with any real non-academic problems that could use work in optimization or stochastic processes.

When I was a B-school prof, I was shocked to discover that B-schools were fine with some quite pure math research but wanted nothing to do with education or practice as in the law school, medical school, school of pharmacy, or agricultural college.

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