By Michael C. Gemignani

Topology is among the such a lot swiftly increasing parts of mathematical idea: whereas its roots are in geometry and research, topology now serves as a strong software in nearly each sphere of mathematical research. This ebook is meant as a primary textual content in topology, available to readers with not less than 3 semesters of a calculus and analytic geometry sequence.

In addition to fabulous assurance of the basics of metric areas, topologies, convergence, compactness, connectedness, homotopy idea, and different necessities, *Elementary Topology* offers additional point of view because the writer demonstrates how summary topological notions constructed from classical arithmetic. For this moment version, a variety of routines were additional in addition to a piece facing paracompactness and whole regularity. The Appendix on countless items has been prolonged to incorporate the final Tychonoff theorem; an evidence of the Tychonoff theorem which doesn't depend upon the speculation of convergence has additionally been extra in bankruptcy 7.

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It is not, however, the topology induced on R 2 by the Pythagorean metric D; for a subset of R 2 can exclude at most finitely many lines and still be open in the topology determined by but {(x, y) \x2 + y2 < 1} ex cludes infinitely many lines and still is D-open. 2 Bases and Subbases 43 EXERCISES 1. Prove that the intersection of finitely many open sets is open and that the union of finitely many closed sets is closed. 2. Suppose X, t a topological space, and A C l . The interior of A, denoted by A°, is defined by A° = U {U E r | U C A}.

C) Prove or disprove: {H\{x> y) | (x, y) E X X X, x j* y} is a subbasis for the topology induced on X by D. 3 OPEN NEIGHBORHOOD SYSTEMS Although we already have four ways to specify a topology on a set, we have not as yet formally introduced one of the most widely used manners of determining a topology. Actually, however, we have already en countered this method, since it is nothing more than the generalization of the p-neighborhoods of a point in a metric space. Definition 5. Suppose X, r to be a topological space, and suppose that for each point x G X we have a collection 91* of open sets having the following properties: i) 91* ^ 0.

Since, for x we have found such that N(x, q) C f ~ l (U), then/ '{(7) is open. Suppose, on the other hand, that is an open subset of A' when ever U is an open subset of Y. We have previously shown that if /(a) e Y and if pis any positive number, then N(f(a), p) is an open subset of Y; therefore f~ 1(N(J(a), p)) is an open subset of X. There is therefore a positive number q such that N(a, q) c r l (N(J(a),p)). We have then that for this q, f(N(a, q)) C p ) ; hence / is continuous. 6 31 The following proposition is quite similar to Proposition 8, but is often easier to apply.