chapter 3 quotient spaces of topological spaces

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Chapter 3 23 Quotient Spaces of Topological Spaces Let X be a vector (or linear) space over the field K and E be a subspace of X. For any x X, the set x + E = { x + e : e X } is called coset of the element x with respect to E. We shall denote x + E by xˆ . 3.1.8 For the subspace E of the linear space X over the field K, the following statements are equivalent : (i) u x + E (ii) u – x E (iii) x u + E 3.1.9 Let E be a subspace of a linear space X over the field K. Then x + E = y + E x – y E and thus (x + z) + E = x + E for any z E 3.1.10 The distinct cosets of the subspace E of a linear space X over the field K form a partition of X. The result 3.1.9 asserts that there is an equivalence relation “” in X such that the distinct cosets of E are the equivalence classes with respect to “”. Thus if xˆ is one of the distinct cosets of E then xˆ is an equivalence class with x as its representative element. If x , y X be such that xˆ = yˆ then by 3.1.8, we have x – y E. Two elements x , y of X are said to be equivalent modulo E if x – y E. This will be denoted by x y(mod E). The set of all the equivalence classes modulo E is denoted by X/E. Thus X/E = { xˆ : x X}.
Chapter 3 24 Quotient Spaces of Topological Spaces The set X/E can be made into a linear space over the field K under algebraic operations defined by (x + E) + (y + E) = (x + y) + E and (x + E) = x + E The linear space X/E over the field K is called the quotient space of X modulo E. elementary. The following results regarding the quotient space of X modulo E are 3.1.11 dim(X/E) = dim X – dim E ; 3.1.12 The quotient mapping Q : X X/E defined by Q(x) = x + E, is a linear mapping. 3.1.13 The null space (or kernel) of Q : X X/E is exactly the subspace E. 3.1.14 Let E1 and E2 be the subspaces of a linear space X over the field K. E2 is a complementary subspace of E1, that is, X = E1 E2 if and only if the restriction of Q to E2 is an isomorphism of E2 onto X/E1. 3.1.15 Let X1 and X2 be linear spaces over the field K. Suppose that f is a linear mapping of X1 onto X2. If N is the null space of f then X2 is isomorphic to X1/N. 3.1.16 Let M be a subspace of the linear space X. If M o represents annihilator of M, then (a) M o (X/M)* (b) M* X*/M o
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chapter 3 quotient spaces of topological spaces

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