By Donald W. Kahn
First-class textual content bargains accomplished assurance of effortless basic topology in addition to algebraic topology, particularly 2-manifolds, protecting areas and basic teams. The textual content is obtainable to scholars on the complicated undergraduate or graduate point who're conversant with the fundamentals of genuine research or complex calculus. difficulties, with chosen options. Bibliography. 1975 version.
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Extra info for Topology: An Introduction to the Point-Set and Algebraic Areas
Sample text
We call B(x, r) the open ball with center x and radius r. Likewise B(x, r) = {t ∈ X : d(x, t) ≤ r} is the closed ball with center x and radius r. 9 The reader may verify as an exercise that, in an arbitary metric space, the closed ball B(x, r) is not necessarily equal to the closure of the open ball B(x, r). However, it will contain that closure. 10 Let (X, d) be a metric space. A set U ⊆ X is said to be open if, for each u ∈ U, there is an > 0 such that B(u, ) ⊆ U. 44 CHAPTER 1. FUNDAMENTALS It is easy to check that the open sets U specified in the last definition form a topology on X in the usual sense.
We see, then, that the Sorgenfrey topology contains all the usual open sets and some new ones as well. We conclude then that (−∞, 0) and [0, ∞) are both open. And certainly R = (−∞, 0) ∪ [0, ∞). So R is disconnected in the Sorgenfrey topology. 5 The topologist’s sine curve (another famous example) is the set 1 S = {(0, y) : y ∈ R} x, sin :x>0 . x 30 CHAPTER 1. 13: The topologist’s sine curve. 13. It is connected. For certainly the left-hand portion of S, which is the yaxis, is connected. And any open set that contains that portion will contain a neighborhood of the origin and hence intersect the right-hand portion (which gets arbitrarily close to the origin).
Thus they cannot be connected to P . So the entire ball is in the complement of S. We conclude that the complement of S is open hence S is closed. Since S is nonempty, open, and closed, we conclude by the connectedness of Euclidean space that S is all of space. That gives the desired conclusion. 6 Prove that the continuous image of a path-connected space is path-connected. A topological space is said to be locally path-connected if it has a neighborhood basis consisting of path-connected sets. 1 for the concept of neighborhood basis.