By John B. Conway

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in element set topology is aimed toward an upper-undergraduate viewers. Its mild velocity could be worthwhile to scholars who're nonetheless studying to jot down proofs. must haves contain calculus and not less than one semester of research, the place the coed has been safely uncovered to the information of easy set idea equivalent to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the actual line. Appendices are integrated to bridge the space among this new fabric and fabric present in an research direction. Metric areas are one of many extra favourite topological areas utilized in different components and are for this reason brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the strategy of the booklet first of all the actual and paintings towards the extra basic. 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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5. 6). Since a ∈ A and b ∈ B, neither is empty, contradicting the assumption that X is connected. 3. The continuous image of a connected set is connected. Proof. Let f : (X, d) → (Z, ρ) be a continuous function and E a connected subset of X; we want to show that f (E) is a connected subset of Z. By replacing X with E, we may assume X = E is connected; by replacing Z with f (E), we may assume that f is surjective. We must now show that Z is connected. If D is a subset of Z that is both open and closed, then the continuity of f implies f −1 (D) is both open and closed in X.

If y ∈ G, then there is an ϵ > 0 such that there is an ϵ-chain x1 , . . , xn in G from x to y. 6(b) says B = k=1 B(xk ; ϵ) is connected. Condition (i) of the definition of an ϵ-chain implies B ⊆ G, and so B ⊆ H. In particular, y ∈ H. Since y was arbitrary, H = G and G is connected. Now assume that G is connected. Fix a point x in G, and let D = {y ∈ G : there is an ϵ > 0 and an ϵ-chain in G from x to y} The strategy of the proof will be to show that D is both relatively open and closed in G; since it is not empty (x ∈ D), it will then follow that D = G, and so G will have been shown to satisfy the condition.

A) Give three examples of σ-compact metric spaces that are not compact. (b) Show that a σ-compact metric space is separable. 5. Connectedness In this section, we introduce and explore another important concept for metric spaces. Consider the following two examples of subsets of R. The first is the set X = [0, 1] ∪ (2, 3), and the second is Y = [0, 1] ∪ (1, 2). In X there are two 30 1. Metric Spaces distinct “parts,” [0, 1] and (2, 3). ) In a sense, writing Y as the union of those two sets is just accidental.

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

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