By John B. Conway

This textbook in aspect set topology is aimed toward an upper-undergraduate viewers. Its mild speed may be important to scholars who're nonetheless studying to put in writing proofs. must haves comprise calculus and at the least one semester of study, the place the coed has been safely uncovered to the information of uncomplicated set thought similar to subsets, unions, intersections, and features, in addition to convergence and different topological notions within the genuine line. Appendices are integrated to bridge the distance among this new fabric and fabric present in an research direction. Metric areas are one of many extra regularly occurring topological areas utilized in different components and are for that reason brought within the first bankruptcy and emphasised through the textual content. This additionally conforms to the technique of the e-book firstly the actual and paintings towards the extra common. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of concept, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact areas.

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Topological Spaces (d) Let {(Xi , Ti ) : i ∈ I} be a collection of topological spaces, where the sets {Xi : i ∈ I} are pairwise disjoint subsets of some larger set that we can take to be X = i∈I Xi . ) Let T = {G ⊆ X : G ∩ Xi ∈ Ti for each i ∈ I}. (X, T ) is a topological space. (e) Let X be any set, and let T be the collection of all subsets G such that X\G is finite. It follows that T is a topology on X called the cofinal topology. We won’t see much of this topology. (f) If (X, T ) is a topological space, Y ⊆ X, and TY ≡ {Y ∩ G : G ∈ T }, then (Y, TY ) is a topological space.

16. 17(a) does not hold as the function f (x) = x2 maps Cauchy sequences to Cauchy sequences, but, as we have seen, it is not uniformly continuous (Exercise 10). Exercises (1) Show that for any number a the function f : [0, 1] → R defined by f (x) = sin(x−1 ) when x ̸= 0 and f (0) = a is not continuous at 0. (2) If (X, d) is a metric space, f : B(a; r) → R is continuous at a with f (a) = 0, and g : B(a; r) → R is a bounded function (but not necessarily continuous), then f g is continuous at a. (3) If f : (X, d) → (Z, ρ) is continuous, A is a dense subset of X, and z ∈ Z such that f (a) = z for every a in A, show that f (x) = z for every x in X.

The following theorem is the main result on compactness in metric spaces. 5. The following statements are equivalent for a closed subset K of a metric space (X, d). (a) K is compact. (b) If F is a collection of closed subsets of K having the FIP, then it holds that F ∈F F ̸= ∅. (c) Every sequence in K has a convergent subsequence. (d) Every infinite subset of K has a limit point. (e) (K, d) is a complete metric space that is totally bounded. Proof. (a) implies (b). Let F be a collection of closed subsets of K having the FIP.