By L. Christine Kinsey
This ebook goals to supply undergraduates with an knowing of geometric topology. themes lined contain a sampling from point-set, geometric, and algebraic topology. The presentation is pragmatic, keeping off the recognized pedagogical process "whereby one starts off with the overall and proceeds to the actual merely after the scholar is simply too careworn to appreciate it." workouts are a vital part of the textual content. scholars taking the direction must have a few wisdom of linear algebra. An appendix presents a short survey of the required historical past of crew thought.
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Extra info for Topology of Surfaces (Undergraduate Texts in Mathematics)
Example text
27. 9. 25 and Exercise We do not intend a thorough study of point-set topology in this text, but only wished to provide a sampling. For more on the intricacies of this subfield of topology, you should consult a text devoted to it, such as James R. Munkres' Topology: A First Course. 4 Product spaces In this section and the next, we investigate ways of creating new topological spaces from old ones. 2 is an example of a product space. 27) Definition. Let X and Y be any spaces. 4 Product spaces 49 Fig.
9) Definition. Let X be a topological space with two bases 'B and 'B'. Let 'J and 'J' be the collections of all open sets corresponding to these bases. Then 'B is equivalent to 'B' if'J = 'J'. 10) Theorem. Let X be a topological space with two bases'B and 'B'. 5. B'. 10. 6. 2, are equivalent. I> Equivalent bases give the same collection of open sets, and so the same closed sets, connected sets, and continuous functions. We can use these bases interchangeably. A different structure on ]Rl is defined by using the half-open topology: Let an interval of the form [a, b) be considered as a neighborhood of any x E [a, b).
Repeat the process to obtain a sequence of smaller and smaller nested cubes each containing infinitely many points of the sequence. For J1 = [-1,1], we let al = -1, b1 = 1. Divide the interval in half, and choose the half with infinitely many of the Xk'S. If both halves have infinitely many, just choose whichever you like best. Then either a2 = -1 and b2 = 0, or a2 = and b2 = 1, depending on your choice. Take the interval [a2' b2 ] and repeat to get ak and bk the endpoints of the kth subinterval selected, Note that b k = ak + 2 i -2 • Thus, limk-+oo bk = limk-+oo ak, assuming that the limits exist.