By Michael C. Gemignani
Topology is without doubt one of the so much speedily increasing parts of mathematical suggestion: whereas its roots are in geometry and research, topology now serves as a strong device in virtually each sphere of mathematical learn. This e-book is meant as a primary textual content in topology, available to readers with a minimum of 3 semesters of a calculus and analytic geometry sequence.
In addition to brilliant insurance of the basics of metric areas, topologies, convergence, compactness, connectedness, homotopy idea, and different necessities, Elementary Topology supplies further point of view because the writer demonstrates how summary topological notions constructed from classical arithmetic. For this moment version, a number of routines were extra in addition to a piece facing paracompactness and whole regularity. The Appendix on limitless items has been prolonged to incorporate the overall Tychonoff theorem; an evidence of the Tychonoff theorem which doesn't rely on the idea of convergence has additionally been extra in bankruptcy 7.
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Extra info for Elementary Topology
Example text
Suppose W is a pubset of 7, D. Prove that the function i : W~ ^ Y defined by i(w) = w for each w G W is continuous as a function from Wy D | W into 7, D. 7. Prove that a function / from a space X, D into a space 7, D' is continuous if and only if given any convergent sequence S in X, f(S) is a convergent se quence in 7. l 8. 3, Exercise 6. Prove that metrics D and D' on a set X are equivalent if and only if the identity map from both X, D onto X, D' and from X, D' onto X, D is continuous. 7 34 9.
Find sets topologically associated with A. 7. Is it possible for two distinct subsets of a topological space to have exactly the same topologically derived sets? Support your assertion. 8. Suppose t and r' are topologies on a set X. Determine if each of the following conditions implies either r C r' or r' C r. In the following, A stands for any subset of X; we use ' to indicate that a derived set is being taken relative to r'. 5. Through out this section X, r will be assumed to be a topological space.
Set p = min (|1 — x\y |x|). Then N(x, p) C R — [0, 1]. Therefore R — [0, 1] is open, and hence [0, 1] is closed (Fig. 9). Example 11. If X is a set with metric D, if x e X, and if p is any positive number, then the closed p-neighborhood of x, denoted by C1N (x, p), is defined to be the set of all y G X such that D(x, y) < p, that is, C1N (x,p) = {y eX | D(x, y) < p}. It is left as an exercise to show that C1N (x, p) is a closed subset of X. Note in Example 10 that [0, 1] = C1N (J, J); therefore the fact that [0, 1] is closed follows from the more general considerations of this example.