You have studied a number of mathematical structures. Vector space, metric space, topo- logical space, group,...

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You have studied a number of mathematical structures. Vectorspace, metric space, topo- logical space, group, ring and field aresome examples. Give general definitions and specific examples.Comment on some of the details of some of these structures. Explainhow various kinds of functions are involved in thesestructures.

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VECTOR SPACE A vector space over consists of a set along with two operations subject to theseconditionsFor any For any For any There is a zero vector such that for all Each has an additive inverse such that If is a scalar that is a member of and then the scalar multiple is in If andthenIf and then If and then For any exampleThe set of realvalued functions of the realvariable is a vector space under theoperationsand inherited from the space in the prior example We can think of as the same as in that corresponds to the vector with and METRIC SPACEA metric space is a set X togetherwith a function d called a metric ordistance function which assigns a real numberdx y to every pair xy X satisfying the properties or axiomsdx y 0 and dx y 0 x ydx y dyxdx y dyz dx zExamplesThe prototype the line R with its usualdistance dx y x yThe plane R2 with the usualdistance measured using Pythagorass theoremdx1 y1x2 y2 x1 x22 y1 y22This is sometimes called the 2metric d2 TOPOLOGICAL SPACELet X be a set A set of subsets of X is called a topologyand the elements of are called open sets if the following properties aresatisfiedthe empty set X if Ai i I then Aiif A B then A B ExamplesThe prototypeLet X be any metric space and take to be the set of open sets as defined earlier The propertiesverified earlier show that is a topologySome extremal examplesTake any set X and let X Then is a topology called See Answer

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