A subset U C E is called open if. for every x e U, there is an open i.pdf

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A subset U C E is called open if. for every x e U, there is an open interval (a, b) where x e (a.b) C U. Show that, in the above definition, the numbers a, b may be taken as rational; that is. if x e U. there is an open interval (c, d) where x (c, d) C U and where c, d e Q. Show that any open set U is a union of (possibly infinitely many) intervals (a. b) where a, b e Q. How many open subsets of E exist? Solution suppose x is any element of U and is any smallestpositive real number. then if ( x - , x + ) is in U , then U isan open set in R. if x is a rational number , then while is a rationalnumber, we have sum and difference of rational numbers isrational. so, x - = a and x + = b such that x always liesbetween x - and x + which can also be written as a< x < b or , x is in ( a,b) . i.e. every element x of U has its neighbourhood inU. so, U is an open set. (b) if U is an open set , then for every x in U , we canchoose such that x is in ( x - , x +). further, is an arbitrarily smallpositive real number and can take infinitely many values , wecan see that there can be infinite intervals in which x lies. further, the union of all those intervals lie inU. thus, union of any number of open intervals is in U. (c0 the set of real numbers is an open set and can be writtenas the union of infinite number of open intervals..

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TOPOLOGY 541 Let M be a set with two members a and b. Define the function D : M × M as follows: D(a,a) = D(b,b) = 0,D(a,b) = D(b,a) = r where r is a positive real number. Prove that (M, D) is a metric space. Solution Definition 1.1 (Metric). A metric, or distance function, on a set X is a mapping d : X × X R such that • d(x, y) 0 for all x, y X, and d(x, y) = 0 if and only if x = y. • d(x, y) = d(y, x) for all x, y X. • d(x, z) d(x, y) + d(y, z) for all x, y, z X. We call (X, d) a metric space. Definition 1.2 (Open ball). Let (X, d) be a metric space. For x X and > 0, the set Bd(x, ) defined by Bd(x, ) = {y X | d(x, y) < } is callen an open ball in the set X. Definition 1.3 (Open sets in metric spaces). Let (X, d) be a metric space and let U be any subset of X. Then U is called an open set in X if every point of U is an interior point of U; that is, for any a U, there is an open ball B(a, ) such that B(a, ) U. Definition 1.4 (Properties of open sets). Let (X, d) be a metric space. • and X are open. • The union of an arbitrary collection of open sets is open. • The intersection of a finite number of open sets is open. Definition 1.5 (Closed set). A subset A of a metric space (X, d) is closed if it’s complement X\\A is open in X. Definition 1.6 (Properties of closed sets). Let (X, d) be a metric space. • and X are closed. • The union of an finite collection of closed sets is closed. • The intersection of an arbitrary number of closed sets is closed. Definition 1.7 (Limit point of a subset). Let (X, d) be a metric space and let A be a subset of X. Then a point x in X is a limit point of A if every open ball B(x, ) contains at least one point of A. The set of all limit points of A is called the derived set A . Definition 1.8 (Closure of a set). Let (X, d) be a metric space and let A X. Then the set consisting of A and its limit points is called the closure of A, denoted A. A = A A Corollary 1.19. We note that A is dense in X if and only if for any x X and > 0, there is a point a A such that d(x, a) < . 1.2. Subspaces. Definition 1.20 (Open sets in a subspace). Let (X, d) be a metric space and (Y, dY ) be a metric subspace of (X, d). Let G be a subset of Y . Then G is open in Y if and only if, for any x G, there is an open ball B(x, ) in X such that B(x, ) Y G A subset H of Y is closed in Y if its complement G = Y \\H of H is open in Y . Theorem 1.21 (Open sets in a metric subspace). Let (Y, dY ) be a metric subspace of a metric space (X, d), and let G Y . Then G is open in Y if and only if there exists an open subset U in X such that G = U Y . 1.3. Convergence in a Metric Space. Definition 1.22 (Convergence). A sequence (xn) in a metric space (X, d) is said to converge to a point x X if for any > 0, there exists N such that n > N implies d(xn, x) < The point x is called a limit of the sequence (xn) Corollary 1.23. A sequence (xn) in a metric space (X, d) is said to converge to a point x X if any open ball B(x, ) contains almost all xn. Theorem 1..

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Find the indicial roots then solve the following ODE using series. You must show the first 4 terms of each series. xy\" + 2y\' + xy = 0 Solution Ans- Let X be a set and let be a family of subsets of X. Then is called a topology on X if:[1][2] If is a topology on X, then the pair (X, ) is called a topological space. The notation X may be used to denote a set X endowed with the particular topology . The members of are called open sets in X. A subset of X is said to be closed if its complement is in (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open. Basis for a topology Main article: Basis (topology) A base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.[3][4] We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them. Subspace and quotient Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : XY is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes. Examples of topological spaces A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must b.

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