Topological Spaces
From Distance to Neighborhood
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This book is a text, not a reference, on Point-set Topology. It addresses itself to the student who is proficient in Calculus and has some experience with mathematical rigor, acquired, e.g., via a course in Advanced Calculus or Linear Algebra. To most beginners, Topology offers a double challenge. In addition to the strangeness of concepts and techniques presented by any new subject, there is an abrupt rise of the level of abstraction. It is a bad idea to teach a student two things at the same moment. To mitigate the culture shock, we move from the special to the general, dividing the book int...
This book is a text, not a reference, on Point-set Topology. It addresses itself to the student who is proficient in Calculus and has some experience with mathematical rigor, acquired, e.g., via a course in Advanced Calculus or Linear Algebra. To most beginners, Topology offers a double challenge. In addition to the strangeness of concepts and techniques presented by any new subject, there is an abrupt rise of the level of abstraction. It is a bad idea to teach a student two things at the same moment. To mitigate the culture shock, we move from the special to the general, dividing the book into three parts: 1. The Line and the Plane 2. Metric Spaces 3. Topological Spaces. In this way, the student has ample time to get acquainted with new ideas while still on familiar territory. Only after that, the transition to a more abstract point of view takes place. Elementary Topology preeminently is a subject with an extensive array of technical terms indicating properties of topological spaces. In the main body of the text, we have purposely restricted our mathematical vocabulary as much as is reasonably possible. Such an enterprise is risky. Doubtlessly, many readers will find us too thrifty. To meet them halfway, in Chapter 18 we briefly introduce and discuss a number of topological properties, but even there we do not touch on paracompactness, complete normality, and extremal disconnectedness-just to mention three terms that are not really esoteric.
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