01 - manifolds - INTRODUCTION TO MANIFOLDS I Definitions...

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Unformatted text preview: INTRODUCTION TO MANIFOLDS I Definitions and examples 1. Topologic spaces Definition. A topological space M is an abstract point set with explicit indication of which subsets of it are to be considered as open. Such open-by-definition subsets are to satisfy the following tree axioms: (1) and M are open, (2) intersection of any finite number of open sets is open, and (3) union of any (infinite) number of open sets is open. After the notion of an open set is introduced, all the remaining notions of an elementary analysis appear: closed sets as complementary to open ones, compact subsets as admitting selection of a finite subcovering from any (infinite) open covering, connected sets as not representable as unions of disjoint open sets, continuous mappings between two topological spaces as mappings yielding open preimages for open subsets of a target space, converging sequences of points as the ones which get into any open set containing the limit, after sufficiently many steps, etc . Examples. Example. Discrete (finite or infinite) sets with the discrete topology: all subsets are declared to be open. Problem 1. Describe all connected discrete spaces, compact discrete spaces. Example. Normal spaces like R n , intervals (open and closed), etc . Problem 2. Why a closed interval [0 , 1] R 1 is a topological space? Example. Zarissky topology on R and Z : open are sets whose complement is finite, and the only other open set is the empty set. Problem 3. Prove that a subset is closed in Zarissky topology, if and only if it is the zero set of a polynomial R R (resp., Z Z ). Example. A set consisting of two elements, a and b , with the following open subsets: ,a, { a,b } . Problem 4. Describe all different types of topological spaces consisting of 3 and 4 points, so that the ones differing only by re-enumeration of points, would Typeset by A M S-T E X 1 2 DEFINITIONS AND EXAMPLES be considered as identical. Is there a formula for the number of non-equivalent (in such a sense) spaces for a general n ? (I dont know the answer). Definition. A base of a topology is a family of open subsets such that any other open set may be represented as the union of subsets constituting the base of the topology. Example. Surgery on normal topological spaces: the line with two zeros. Take two copies of the real line, R 1 and R 2 , and glue them together by all nonzero points. In other words, consider the equivalence relation on R 1 R 2 as follows, x 1 x 2 x 1 = x 2 6 = 0, and look at the quotient space ( R 1 R 2 ) / . Open (by definition) are the sets such that their full prototypes in R 1 R 2 are open....
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This note was uploaded on 08/12/2008 for the course MATH 4540 taught by Professor Protsak during the Spring '08 term at Cornell University (Engineering School).

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01 - manifolds - INTRODUCTION TO MANIFOLDS I Definitions...

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