Scoville, Nicholas A

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Ursinus College

Ursinus College Digital Commons @ Ursinus College Topology Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Summer 2017 The Closure Operation as the Foundation of Topology Nicholas A. Scoville Ursinus College, nscoville@ursinus.edu Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_topology  Part of the Curriculum and Instruction Commons, Educational Methods Commons, Geometry and Topology Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits you. Recommended Citation Scoville, Nicholas A., "The Closure Operation as the Foundation of Topology" (2017). Topology. 4. https://digitalcommons.ursinus.edu/triumphs_topology/4 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Topology by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact aprock@ursinus.edu. The Closure Operation as the Foundation of TopologyNicholas A. Scoville∗April 19, 20211 IntroductionIn the early 1900s, axiomatizing different mathematical disciplines was all the rage. While a disciplelike geometry was well established by that time, topology was still quite new. Hence, different ways toapproach its axiomatization were still being explored. We will study the work of two mathematicianswho contributed to that effort: Felix Hausdroff and Kazimierz Kuratowski.Born in 1868 to Jewish parents, Felix Hausdroff (1868–1942) is known as one of the founders ofmodern topology, but he also made significant contributions to set theory, descriptive set theory,measure theory, function theory, and functional analysis. Hausdorff studied mathematics and as-tronomy mainly in the city of Leipzig. He graduated from the University of Leipzig and became alecturer at the university as well. In 1901, he was appointed as an adjunct assertion in the Universityof Leipzig,where he taught until 1910 when he accepted a position at the University of Bonn. Haus-dorff continued to teach as a university professor, primarily in Bonn, until he was forced to retire bythe Nazi regime in 1935. After being informed that they were to be placed in an internment camp,Hausdorff committed suicide, together with his wife and his wife’s sister, in 1942. His pioneeringcontributions to the field of topology during his lifetime included development of the concept knowntoday as Hausdorff spaces, as well as the concepts of metric and topological spaces. The developmentof the idea of closeness independent of the ability to be measured also interested Hausdroff.A Polish mathematician and logician, Kazimierz Kuratowski (1896–1980) was born in Warsawto a well-known lawyer in 1896. In 1913, he enrolled as an engineering student at the University ofGlasgow, evidently because he did not wish to study in Russian. Two years later, he began studyingmathematics in Poland after the first World War forced him to return to in Poland. In 1933, hebecame a professor at Warsaw University and began his studies on applications of topological spacesin other areas of mathematics. One of his most significant contributions to general topology was anaxiomatization of the closure operator in which he used Boolean algebra to characterize the topologyof an abstract space without relation to the notion of points. This latter contribution is the mainfocus of this project.∗Mathematics and Computer Science; 601 E. Main Street; Ursinus College; Collegeville, PA 19426;nscoville@ursinus.edu.12 Closed Sets and ClosureIn his foundational textbook on topology titled Set Theory [Hausdorff, 1957, p 257-258], Hausdorffwrote:1∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞The mathematical discipline concerned with [topological invariance] is called Topologyor Analysis Situs. (The latter term, due to Leibniz, was re-introduced by Riemann.) . . .This seems like a suitable occasion to touch, in all brevity, on those point-set theories thatemphasize the topological point of view from the very beginning and work only with thetopologically invariant concepts. . . . . . .What are primary in the topological space X are the sets that are closed (in X) and theircomplements, the open sets; . . . . . . The closed or open sets can be taken as our starting pointand left undefined, or they can be defined from related concepts (limit point, neighborhood),but always derived in such a way as to keep invariant their topological character; the moredetailed nature of the space is then determined by axioms . . . . . . .[Closed] Sum and Intersection Axioms. The closed sets must, regardless ofanything else, satisfy the following conditions:1. The space X and the null set ∅ are closed.2. The union of two closed sets is closed.3. The intersection of any number of closed sets is closed.As a consequence, the closure A can be defined as the intersection of all the closed setscontaining A . . . it has the following properties:Closure Axioms.1. ∅ = ∅2. A ⊆ A3. A = A4. A ∪B = A ∪BIt would also be possible to make these properties our starting point (à la Kuratowski) bytaking them as our axioms for the set function A.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞The goal of this project is to prove the last claim made by Hausdorff concerning the possibilityof making Kuratowksi’s four Closure Axioms “our starting point.” We will also show, in Task 6, thatthis axiomatization is equivalent to the “open set axioms” with which you are familiar. In otherwords, we will show that the Open Sum and Intersection Axioms (OSIA), which can be found in any1Hausdorff’s first book-length treatment of topology was entitled Grundzüge der Mengenlehre (Fundamentals of SetTheory) [Hausdorff, 1914], prior to Kuratowski’s work. Later, Hausdorff published a significantly revised version of his1914 textbook — essentially an entirely new book — that appeared in two editions (in 1927 and 1935) under the titleMengenlehre (Set Theory). In this project, we use excerpts from the published English translation, [Hausdorff, 1957],of the last German edition, [Hausdorff, 1935], with minor changes by the project author. For instance, throughoutthis project, we use today’s set notation ∪, ∩ in place of the boolean algebra notation +, × that both Hausdorff andKuratowski employed for the operations of union and intersection respectively.2modern introductory textbook on topology, are equivalent to the Closure Axioms (CA). To set thestage for this work, we begin by examining Hausdorff’s assertion that “the closed or open sets can betaken as our starting point and left undefined” and his definition of closure.Task 1 Prove that Hausdorff’s Closed Sum and Intersection Axioms (CSIA) are equivalent to OSIAthat we use today. [Hint: Use De Morgan’s laws.] When starting with CSIA, “closed” is anundefined term and A ⊆ X is open if X − A is closed. Similarly, when starting with OSIA,“open” is undefined and A ⊆ X is closed if X −A is open.Task 2 Let A be any set, and assume CSIA. Using Hausdorff’s definition of A, prove that A is closed.3 Kuratowski on the Closure AxiomsWe now investigate a very brief selection from a paper by Kuratowski entitled “Sur l’opération Ade l’Analysis Situs (“On the Closure Operation in Topology”), [Kuratowski, 1922]. As Kuratowskidescribed in a footnote to its title, this paper was “the first part — slightly modified — of my thesispresented on 12 May 1920 at the University of Warsaw for the degree of Doctor of Philosophy.” Heexplained the purpose of his paper, as well as the role of the four statements referenced by Hausdorffas the Closure Axioms, as follows:2∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞This paper is dedicated to an analysis of these propositions [CA] and their consequences.We proceed axiomatically: we assume as given an arbitrary3 set X and a function A, suchthat for every subset A of X, there exists a subset A also contained in X and satisfying theaxioms 1–4.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞It is worth pondering exactly what Kuratowski has set up here. As mentioned above, axiomaticsystems were all the rage at this time, and Kuratowski was right in the thick of it. Essentially, hewas declaring that the four stated properties are so fundamental and so robust, that we can decreethat we have an abstract set operation, denoted A, that satisfies them as axioms. From there, wecan investigate the theory of this operation and prove many things within it. With this in mind, weturn back to Kuratowski.2All translations of Kuratowski excerpts in this project were prepared by Mark Bowron, mathematrucker.com, 2010,with minor changes by Janet Heine Barnett, Colorado State University – Pueblo (retired), 2021.3In his 1922 paper, Kuratowski used an arbitrary subset of Euclidean n-space, not an arbitrary set. Because thiswas a transitional period between metric spaces and more general topological spaces, it is not clear whether Kuratowskirealized at the time that he wrote this paper that his axiomatization applies to general topological spaces. We havechosen to drop Kuratowski’s assumption of Euclidean n-space throughout this project, as the mathematics is exactlythe same but the generality is much greater.3∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞§1. General properties of A.We now establish the [. . .] fundamental properties4 of the closure operation.Theorem 1. A ⊆ B implies A ⊆ B.Theorem 2. A ∩B ⊆ A ∩B.Theorem 3. A−B ⊆ A−B.Theorem 4. X = X.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞We let Kuratowski prove Theorem 1 (but using Hausdorff’s numbering of the Closure Axioms forconsistency).∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞Indeed, the inclusion A ⊆ B implies B = A ∪ B, hence B = A ∪B, and by CA 4,B = A ∪B. Thus A ⊆ B.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞Task 3 Prove Theorems 2–4 assuming CA. [Hint: The following identities from set theory may proveuseful in your proof: A ∩B ⊆ A,A ∩B ⊆ B,A ⊆ A ∪B,A ∪B = (A−B) ∪B.]Kuratowksi also stated the following generalization of Theorem 2.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞Theorem 2a. Letting {Ai} denote an arbitrary family of sets indexed by the variable i, we have∩iAi ⊆∩iAi and∪iAi ⊆∪iAi.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞Task 4 Prove Theorem 2a.4Kuratowksi stated and proved six fundamental properties in his paper, two of which we have omitted since theyare not relevant to the goals of this project.4After proving several other generalized properties of the closure operation, Kuratowski turnedhis attention to other topological concepts.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞§2. Fundamental notions of topologyIn this section we will define some notions of topology, based on the operation A.A set A is said to be closed when A = A.. . . the union of two closed sets is closed; . . . the intersection of any collection of closedsets is closed.∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞Task 5 Kuratowski has claimed here that his definition of closed set satisfies CSIA2 and CSIA3. Usingwhat you and Kuratowski have shown so far, prove this claim.Before turning to our main result, it may help to think about the following. We have two sets ofaxioms (CSIA and CA) and two terms (closed and closure). If we assume the CSIA, which of thetwo terms is undefined and which one needs a definition? If we assume the CA, which of the twoterms is undefined and which one needs a definition? Both definitions are located somewhere in thisproject. Make sure you locate them and have them on hand for the final question.Task 6 Finally, prove that a collection of sets satisfies CA if and only if it satisfies OSIA.ReferencesF. Hausdorff. Grundzüge der Mengenlehre (Fundamentals of Set Theory). Von Veit, Leipzig, 1914.F. Hausdorff. Mengenlehre (Set Theory). von Veit, Leipzig, second edition, 1935. Often identified asthe third edition of [Hausdorff, 1914].F. Hausdorff. Set Theory. Chelsea Publishing Company, New York, 1957. English translation of[Hausdorff, 1935] by John R. Aumann et al.K. Kuratowski. Sur l’opération A de l’Analysis Situs (On the Closure Operation in Topology). Fund.Math., 3:182–199, 1922.5Notes to InstructorsPSP Content: Topics and GoalsThis Primary Source Project (PSP) is for students who have seen the standard “open set axioms” andprovides an equivalent axiomatic system for grounding all of point-set topology based on the work ofKuratowski. Other than exposure to and familiarity with the open set axioms, this is a self containedproject which can easily be done during a single class period with any unfinished tasks assigned forhomework. Because most of the tasks have several claims to prove, this allows the instructor theability to show one claim, another claim to be worked on in groups during class, and other claimsto be done as homework tasks. In addition to the overarching goals of familiarizing students withan axiomatic system and showing them that multiple systems are equivalent, this project has thefurther content goal of familiarizing the student with the closure operation.The closure axioms are not a standard topic in a first course in topology. As such, the projectprovides some interesting opportunities. For example, I have used this project as part of a finalexam in a topology course. Another option is to use this project as a jumping off point to delvedeeper into the nature of axiomatic systems. Different axiomatic systems are a huge topic in acourse in geometry, but it is given little attention in other courses. This project allows one to seethat there are multiple sets of axioms that are equivalent, yet look very different. This raises thequestion as to whether or not one could come up with a different, non-equivalent set of axioms to do“topology”? Just as there are different geometries based on the axioms one chooses, can one createdifferent notions of topologies based on the choice of axioms? What if one of the closure axioms waseliminated or changed? What if another axiom was added? These can be sources of exploration.Student PrerequisitesThis project can be done without any prior knowledge of topology. In that sense, this project couldbe a project on day 1 of a topology course. However, it is recommended that students have beenexposed to come concepts in topology, particularly that they have some familiarity with the open setaxioms. The reason is because without that context, the material in this PSP will seem unmotivatedand abstract. The fact that the closure operation axioms are equivalent to the open and closed setaxioms will be appreciated more if students are aware of the fact that all of topology is built up fromthe open set axioms.PSP Design, and Task CommentaryBecause the nature of this project is axiomatic, the tasks are fairly straightforward and clean. Task1, for example, is a simple application of deMorgan’s laws and a standard exercise in any firstcourse in topology. LATEX code of the entire PSP is available from the author by request to facilitatepreparation of advanced preparation / reading guides or ‘in-class worksheets’ based on tasks includedin the project. The PSP itself can also be modified by instructors as desired to better suit their goalsfor the course.Suggestions for Classroom Implementation and Sample ScheduleThis project can easily be completed in two 50-minute class periods, possibly even one. Assignreading up to Task 2 and completion of the first two tasks as advance preparation for the first day.Class can then begin with a discussion about the reading, the nature of the problem that is underconsideration, and the nature of axioms. Students can share their solutions for the first two tasks on6the board, and a whole-class discussion can ensue to ensure that students understand what was donein the first task and why it is important. Students can be put into small groups to spend the rest ofthe class period working on the remaining exercises. These can either be written up and handed inthe next day, or the class can go over them together the next day.Connections to other Primary Source ProjectsThe following additional primary source-based projects by the author are also freely available for usein teaching courses in point-set topology. The first three projects listed are full-length PSPs thatrequire 10, 5 and 3 class periods respectively to complete. All others are designed for completion in2 class periods.• Nearness without Distance• Connectedness: Its Evolution and Applications• From Sets to Metric Spaces to Topological Spaces• Topology from Analysis (Also suitable for use in Introductory Analysis courses.)• The Cantor set before Cantor (Also suitable for use in Introductory Analysis courses.)• Connecting Connectedness• A Compact Introduction to a Generalized Extreme Value TheoremClassroom-ready versions of these projects can be downloaded from https://digitalcommons.ursinus.edu/triumphs_topology. They can also be obtained (along with their LATEX code) fromthe author.AcknowledgmentsThe development of this student project has been partially supported by the TRansforming Instruc-tion in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Program withfunding from the National Science Foundation’s Improving Undergraduate STEM Education Pro-gram under Grant Nos. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Anyopinions, findings, and conclusions or recommendations expressed in this project are those of theauthor and do not necessarily represent the views of the National Science Foundation.This work is licensed under a Creative Com-mons Attribution-ShareAlike 4.0 International License(https://creativecommons.org/licenses/by-sa/4.0/legalcode).It allows re-distribution and re-use of a licensed work on the conditionsthat the creator is appropriately credited and that any derivative workis made available under “the same, similar or a compatible license.”For more information about TRIUMPHS, visit https://blogs.ursinus.edu/triumphs/.7

The Closure Operation as the Foundation of Topology

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The Closure Operation as the Foundation of Topology

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