By Ethan D. Bloch
The individuality of this article in combining geometric topology and differential geometry lies in its unifying thread: the suggestion of a floor. With a number of illustrations, workouts and examples, the coed involves comprehend the connection among glossy axiomatic procedure and geometric instinct. The textual content is saved at a concrete point, 'motivational' in nature, heading off abstractions. a couple of intuitively attractive definitions and theorems bearing on surfaces within the topological, polyhedral, and tender instances are offered from the geometric view, and aspect set topology is specific to subsets of Euclidean areas. The therapy of differential geometry is classical, facing surfaces in R3 . the cloth here's available to math majors on the junior/senior point.
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Extra resources for A First Course in Geometric Topology and Differential Geometry
Let C C B be closed. Then B - C is open in B, and so f -' (B - C) is open by hypothesis. Using standard properties of inverse images, we have f-'(B-C)= f -'(B) - f -'(C) = A - f-'(C). Since A - f -' (C) is open, it follows that f -' (C) is closed. We have thus proved one of the implications in the lemma. The other implication is proved similarly. 0 We now show that the above definition of continuity in terms of open sets is equivalent to the e-3 definition from real analysis, given in part (3) of the following proposition.
The following are equivalent: (1) A is connected; (2) A cannot be expressed as the union of two non-empty disjoint subsets each of which is closed in A; (3) the only subsets of A that are both open and closed in A are 0 and A. Proof. 1. O The following theorem shows that not only are intervals in R connected, but that they are the only connected subsets of R. The proof of this theorem makes crucial use of the Least Upper Bound Property of the real numbers; consult [HM, p. 38], or most introductory real analysis texts, for a discussion of this property.
Is the property of being disconnected preserved by continuous maps? 6*. 5. 7*. Show that the following sets are path connected, and hence connected: (1) any open ball in IR", and any closed ball in R"; 34 I. Topology of Subsets of Euclidean Space (2) any open ball in R" from which a point has been removed, when n > 2; (3) the unit circle S' C R2. 8. Let U C R2 be an open, path connected set, and let x E U be a point. Show that U - (x ) is path connected. Find an example to show that the hypothesis of openness cannot be dropped.