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LectureNotes2G - Math 528 Geometry and Topology II Fall...

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Math 528 Jan 11, 2005 1 Geometry and Topology II Fall 2005, USC Lecture Notes 2 1.4 Definition of Manifolds By a basis for a topological space ( X, T ), we mean a subset B of T such that for any U T and any x U there exists a V B such that x V and V U . Exercise 1.4.1. Let Q denote the set of rational numbers. Show that { B n 1 /m ( x ) | x Q n and m = 1 , 2 , 3 , . . . } forms a basis for R n . In particular, R n has a countable basis. So does any subset of R n with the subspace topology. Exercise 1.4.2. Let T be the topology on R generated as follows. We say that a subset U of R is open if for every x U , there exist a , b R such that x [ a, b ) and [ a, b ) U . Show that T does not have a countable basis. ( Hint: Let B be a basis for T , and for each x R , let B x be the basis element such that x B x and B x [ x, x + 1).) A toplogical space X is said to be Hausdorf , if for every pair of distinct points p 1 , p 2 X , there is a pair of disjoint open subsets U 1 , U 2 such that p 1 U 1 and p 2 U 2 . Exercise 1.4.3. Show that any compact subset of a Hausdorf space X is closed in X . Exercise 1.4.4. Let X be compact, Y be Hausdorf, and f : X Y be a continuous one-to-one map. Then f is a homeomorphism between X and f ( X ). We say that X R n is convex if for every x , y X , the line segment λx + (1 λ ) y, λ [0 , 1] lies in X Exercise 1.4.5 (Topology of Convex Sets). Show that every compact convex subset of R n , which contains an open subset of R n , is homeomorphic to B n 1 ( o ). ( Hint: Suppose that o lies in the open set which lies in X . Define f : S n 1 R by f ( u ) := sup x X u, x . Show that g : X B n 1 ( o ), given by g ( x ) := x/f ( x/ x ), if x = o , and g ( o ) := o , is a homeomorphism.) 1 Last revised: January 12, 2005 1
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By a neighborhood of a point x of a topological space X we mean an open subset of X which contains x . We say a topological space X is locally homeomorphic to a topological space Y if each x X has a neighborhood which is homeomorphic to Y . By a manifold M , we mean a topological space which satisfies the following properties: 1. M is hausdorf. 2. M has a countable basis. 3. M is locally homeomorphic to R n . The “ n ” in item 3 in the above defintion is called the dimension of M . Exercise 1.4.6. Show that condition 3 in the definition of manifold may be replaced by the following (weaker) condition: 3’. For every point p of M there exist an open set U R n and a one-to-one continuous mapping f : U M , such that p f ( U ). Conditions 1 and 2 are not redondant, as demonstrated in the following Exercise: Exercise 1.4.7. Let X be the union of the lines y = 1 and y = 1 in R 2 , and P be the partition of X consisting of all the subsets of the form { ( x, 1) } and { ( x, 1) } where x 0, and all sets of the form { ( x, 1) , ( x, 1) } where x < 0. Show that X is locally homeomorphic to R but is not hausdorf.
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