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LectureNotes8U - Math 497C Curves and Surfaces Fall 2004...

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Math 497C Oct 8, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 8 2 Surfaces 2.1 Definition of a regular embedded surface An n -dimensional open ball of radius r centered at p is defined by B n r ( p ) := { x R n | dist( x, p ) < r } . We say a subset U R n is open if for each p in U there exists an > 0 such that B n ( p ) U . Let A R n be an arbitrary subset, and U A . We say that U is open in A if there exists an open set V R n such that U = A V . A mapping f : A B between arbitrary subsets of R n is said to be continuous if for every open set U B , f 1 ( U ) is open is A . Intuitively, we may think of a continuous map as one which sends nearby points to nearby points: Exercise 1. Let A , B R n be arbitrary subsets, f : A B be a continuous map, and p A . Show that for every > 0, there exists a δ > 0 such that whenever dist( x, p ) < δ , then dist( f ( x ) , f ( p )) < . Two subsets A , B R n are said to be homeomorphic , or topologically equivalent, if there exists a mapping f : A B such that f is one-to-one, onto, continuous, and has a continuous inverse. Such a mapping is called a homeomorphism . We say a subset M R 3 is an embedded surface if every point in M has an open neighborhood in M which is homeomorphic to an open subset of R 2 . Exercise 2. (Stereographic projection) Show that the standard sphere S 2 := { p R 3 | p = 1 } is an embedded surface ( Hint : Show that the stereographic projection π + form the north pole gives a homeomorphism between R 2 and S 2 (0 , 0 , 1). Similarly, the stereographic projection π 1
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