Morgan, Kyle A's Topological Disquisition

Article: Advanced Plane Topology from an Elementary Standpoint by Donald E. Sanderson in Mathematics Magazine
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Article Summary

The author would like to welcome you to reading his paper about an article the author read, entitled “Advanced Plane Topology from an Elementary Standpoint by Donald E. Sanderson in Mathematics Magazine volume 53 circa 1980 C.E. The author chose this particular article because it has to do with topology and it is purportedly easy to understand, which sounds glorious because the author is not very skilled in mathematics or its lingo.
Donald Sanderson makes a gallant attempt at getting us to understand some fundamentals of topology by relating them to familiar things and even including some pictures. The author of this paper learned a modicum of the Jordan Curve Theorem in his Real Analysis class, and Donald Sanderson reinforced the author’s schema concerning the theorem by relating it to Green’s Theorem, which has to do with a closed curve surrounding a measurable area. Sanderson also briefly relates the all-too familiar Intermediate Value Theorem with the opaque Brouwer Fixed Point Theorem in the way that a point is related to a line. Sanderson was also kind enough to proclaim that the proofs in the article are not so important for students like the author, as long as they get a little bit of understanding.
The Alexander Addition Theorem is a very prominent piece of knowledge in this article. There is a formal proof of the theorem that the author does not understand too well, but there is a picture that shows a path going from one side of a box to another by moving around obstacles that make a sort of maze-like corridor for the path. Even more lucky for the author, Sanderson gives an understandable definition of the Alexander Addition Theorem, stating “that if a path from x to y missing one closed set can be continuously deformed into one missing a second closed set without ever touching a point in both sets, then some path from x to y misses both sets” (82). The author understands the gist of it pretty well, and can also believe Donald Sanderson when he describes how this theorem can apply to all dimensions.
Sanderson spends most of the rest of his article discussing connectivity. He comes back in a circular loop to the Jordan Curve Theorem, where he informs us that Camille Jordan published a proof for the theorem in 1887, but it was not even a legitimate proof at all. We still call it the Jordan Curve Theorem even though Oswald Veblen was the first to prove it in 1905. The very end of the document shows the series of “topological pretzel” pictures, amazingly twisting to hook or unhook at the top. The author thought the article to be sensational.

Connection to Real Analysis

This article is extremely interesting because the author finds it fun to think about things twisting and morphing in various dimensions. The material presented in the Sanderson article are also important because it is directly applicable to the author’s current mathematical education in vector calculus and real analysis. In fact, a very large portion of the text for the author’s real analysis course is centered explicitly around topology. Since the author is not doing so well in both of those classes, he takes any opportunity to gain even a little intuition about the subject matter to be a golden occasion.

Broader Context

The author does not know all too much about the broad field of mathematics, but he could go out on a limb and say that it might apply to the wondrous field of geodesy, which is his favorite subject. We are talking about things that are closed and connected, and then dividing their parts up into smaller parts. Geodesy means to divide the earth, which just happens to be a sphere, and so has 0-connectedness. Topology has a lot of roots in physical shapes like the Klein Bottle or Moebius strip which open great mathematical viewpoints and have strange applications in the real world. Similarly, geodesy can deal with just about any physical shape we desire, and their application can be found in areas such as medicine and architecture.
The reader could go deeper into topology or geodesy by having some no-pressure fun with it. Making a simple Moebius strip out of paper, making an 8V geodesic dome from scratch, or crumpling a numbered piece of paper over a similar flat paper to demonstrate the Brouwer Fixed Point Theorem are all good starting points for learning more in the future. The author is still at this starting point, unfortunately. Day by day, we can all continue gaining a fuller, richer understanding of math.

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