Mathematics is for everyone — whether as a gateway to other fields or as background for higher level mathematics. I made this textbook available to everyone for free download for their own non-commercial use. It could be especially useful for instructors who are looking for an inquiry-based textbook, as a supplemental resource to accompany their course, or for someone interested in learning some topology on their own. If an instructor would like to make changes to any of the files to better suit their students’ needs, source files for the text are available by making a request to the author.

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Over many years I have taught topology I developed pre- and in-class activities that I used to supplement the texts I had adopted. Eventually I had enough material to eliminate the reliance on outside texts and could rely on the activities. This book is built on those activities and is intended as a one semester introduction to point-set topology. The emphasis for this book is to have students be active learners and to help develop their intuition through working activities and examples. Although it is difficult to capture the essence of active learning in a textbook, this book is an attempt to do just that.

Each section contains a collection of exercises. The exercises occur at a variety of levels of difficulty and most force students to extend their knowledge in different ways. While there are some standard, classic problems that are included in the exercises, many problems are open ended and expect a student to develop and then verify conjectures.

This text begins by formally introducing sets and functions. Although these topics are familiar to students, it is my experience that the level of understanding of sets and functions for most students is not yet sufficiently deep enough to ensure success with functions throughout the course. Chapter 2 focuses on metric spaces. It is my belief and my experience that students better understand the abstract concepts of neighborhoods, open sets, continuity, etc., if they are first experienced in a more familiar, concrete context like metric spaces. Metric spaces are easier for students to grasp than general topological spaces as they provide a notion of distance that is comforting to students. Metric spaces also allow one to introduce and motivate important topological concepts in a more familiar context. For example, by first encountering continuity of functions in a metric space setting, and revisiting the idea from different perspectives, the definition of continuity in the more abstract setting of topological spaces seems more accessible and natural. This perspective was also noted by Felix Hausdorff, who is considered as one of the founders of modern topology. His text Grundzüge der Mengenlehre (Fundamentals of Set Theory) (Felix Hausdorff, Leipzig, Von Veit, 1914. Translated to English as Set Theory by John R. Aumann et al, New York, Chelsea PublishingCompany, 1957) provided one of the first systematic treatments of topological spaces. As Hausdorff writes (p. 210) concerning the introduction of the concepts of topology following topics of basic set theory,

While this text is intended as a one semester introduction to point-set topology, there is more material in the text than my students have been able to comfortably digest in a single semester. With that in mind, some care should be taken to ensure that what you feel are the most important topics are ones that are discussed. As mentioned earlier, I think students really benefit from a thorough review and discussion of sets and functions in the first two sections. From there, the sections build on each other. However, one can omit Chapter 4 on Applications of Metric Spaces without loss of continuity. My goal for a single semester course is to be sure that we can investigate compactness and connectedness. With this in mind, it is possible to omit Chapter 16 on quotient spaces if time is tight.

The Grand Valley Steve University libraries provided a grant to support my colleagues, John Golden and Clark Wells, to review a draft version of this text. Their thorough reading and comments on the draft have made for a much better book. I am also indebted to my students who have all been test subjects for earlier versions of this material. Without their hard work and suggestions, this book would not exist.

The HTML version of the text is possible because of the work of Rob Beezer and the PreTeXt team for the development of the PreTeXt platform. David Farmer at the American Institute of Mathematics provided an initial PreTeXt conversion. From this version, Ian Curtis, Editorial Assistant for the GVSU Libraries, as part of the Accelerating Open Educational Resources Initiative at Grand Valley State University², invested a significant amount of time and effort to create the PreTeXt version that you see. This work was conducted with support from the University Libraries and the GVSU President’s Innovation Fund.

The code used to generate this book can be found in the GVSU OER GitHub repository³. This includes the PreTeXt XML files, figures, customization files (XSL), and the output (HTML and PDF). Contributors are encouraged to submit an issue or a pull request with changes, especially regarding accessibility (as mentioned below in Accessibility).

In creating this PreTeXt version we (the author and the Grand Valley State University Libraries) strive to ensure our tools, devices, services, and environments are available to and usable by as many people as possible.

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I have endeavored to keep the prerequisite material to a minimum for this text. The first two sections discuss sets and functions. Most of this material should be review for the reader, but the importance of these ideas in the subsequent sections makes it a good idea to spend time on these sections. That being said, it will be very helpful for the reader to have some background in reading and writing mathematical proofs.

The objectives of this book and its inquiry-based format place the responsibility of learning the material where it belongs — on your shoulders. It is imperative that you engage in the material by completing the preview activities and the in-class activities in order to develop your intuition and understanding of the material. If you do this, and ask questions when you have them, your probability of success will be greatly enhanced. Good luck!