By John B. Conway

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in element set topology is aimed toward an upper-undergraduate viewers. Its light speed could be beneficial to scholars who're nonetheless studying to put in writing proofs. necessities comprise calculus and a minimum of one semester of research, the place the scholar has been safely uncovered to the guidelines of simple set idea similar to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the actual line. Appendices are incorporated to bridge the distance among this new fabric and fabric present in an research direction. Metric areas are one of many extra time-honored topological areas utilized in different parts and are accordingly brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the method of the e-book to begin with the actual and paintings towards the extra basic. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of proposal, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued features, culminating in a improvement of paracompact areas.

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**Additional resources for A Course in Point Set Topology (Undergraduate Texts in Mathematics)**

**Example text**

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. TY is called the subspace topology or relative topology defined by T on Y . We note that this is consistent with what we did when discussing subspaces of a metric space.

If you stick with mathematics a little longer, you will see such things as the Principle of Uniform Boundedness in functional analysis whose proof is a standard application of the Baire Category Theorem. 6, however, when we explore abstract topological spaces in the next chapter. 1 (Baire7 Category Theorem). If (X, d) is a complete metric space and {Un } is a sequence of open subsets of X each of which is dense, ∞ then n=1 Un is dense. Proof. To show that ∞ n=1 Un is dense, it suﬃces to show that if G is a nonempty open subset of X, then G ∩ ∞ n=1 Un ̸= ∅.

7 says that if (X, d) is a metric space and C(X) is the set of all continuous functions f : X → R, then C(X) is a vector space over R. Show that C(X) is a finite-dimensional vector space if and only if X is a finite set. ) (14) Let (X, d) be a metric space, and let U denote the set of all uniformly continuous functions from X into R. (a) If f, g ∈ U and we define (f + g) : X → R by (f + g)(x) = f (x) + g(x) for all x in X, show that f + g ∈ U. In words, U is a vector space over R. (b) If f, g ∈ U and we define (f g) : X → R by (f g)(x) = f (x)g(x) for all x in X, 24 1.

### A Course in Point Set Topology (Undergraduate Texts in Mathematics) by John B. Conway

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