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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**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.