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Title: | MATH 431-01, Topology, Spring 2012 |

Authors: | Seaton, Christopher |

Keywords: | Syllabus;Curriculum;Academic departments;Text;Mathematics and Computer Science, Department of;2012 Spring |

Issue Date: | 11-Jan-2012 |

Publisher: | Memphis, Tenn. : Rhodes College |

Series/Report no.: | Syllabi CRN;22457 |

Abstract: | In Calculus, you made frequent use of results such as the Intermediate Value Theorem, the Extreme Value Theorem, etc., that state properties of continuous functions on an interval [a, b]. Since taking Calculus I, you have met functions with a variety of different sets as domains and ranges. Some of these sets may seem similar to the real numbers or a closed interval of real numbers in one way or another. Hence, it makes sense to ask: for which of these functions do results such as the Intermediate Value Theorem and the Extreme Value Theorem hold? The area of Topology is the mathematical formulation of these questions and their answers. In topology, sets such as the real numbers are stripped of their “unnecessary” structures that do not play a fundamental role in these questions. For instance, the real numbers come equipped with several operations (addition, multiplication, etc.), a linear order, the structure of a real, 1-dimensional vector space, etc. However, many of these structures can be forgotten or simplified, and the remaining object is still sufficiently rich in order to prove the Intermediate Value Theorem and the Extreme Value Theorem. What remains is a topological space, which we can think of as a set with “just enough” structure to define what it means for a function from that space to be continuous. You may have heard topology described as “a kind of geometry where you are allowed to stretch and deform objects.” As is the case for most intuitive descriptions of mathematical concepts, this is both helpful and misleading. It is true that topological spaces cannot be distinguished if they are the same “up to stretching and deforming,” but the reason for this is that stretching an object doesn’t affect whether functions from that object are continuous. Another way of thinking about topology is the study of continuity. And in order to study continuity rather than continuity of functions from the real numbers, continuity of functions from a field, continuity of functions in the presence of other structures, etc., topology considers only those features of a space that are required in order to make sense of continuity. Rather surprisingly, you don’t need to know very much about a space in order to decide when functions are continuous. You just need to know which subsets of the space are considered open. So a topological space is nothing more than a set equipped with a collection of subsets that are called “open sets.” At this level of abstraction, we will develop many familiar properties of continuous functions and spaces as well as a number of surprising, counterintuitive examples. We will learn to ask and answer questions about exactly what properties are required for spaces and continuous functions to exhibit certain behaviors. |

Description: | This syllabus was submitted to the Office of Academic Affairs by the course instructor. Uploaded by Archives RSA Josephine Hill. |

URI: | http://hdl.handle.net/10267/15848 |

Appears in Collections: | Course Syllabi |

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syllabus-M431-01-spring12.pdf | 83.49 kB | Adobe PDF | View/Open |

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