chapter 3 quotient spaces of topological spaces

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Chapter 3 21 Quotient Spaces of Topological Spaces topology for X/R rendering Q continuous on X is called the quotient topology and X/R with the quotient topology is called the quotient space of the topological space X . We shall denote the quotient topology on X/R by Q. Notice that if A is a subset of X then R[A] = Q -1 (Q(A)) and if A X/R, then Q -1 (A) = { A : A A }. 3.1.1 Let Q : (X, ) (X/R, Q) be the quotient map of the topological space X onto the quotient space X/R. Then the following statements are equivalent. (i) Q is an open(closed) mapping. (ii) If G is open(closed) in X, then R[G] is open(respectively closed). (iii) If F is a closed(open) subset of X then the union of all members of X/R which are subsets of F is closed(respectively open). 3.1.2 Let (X, ) be a topological space. If X is compact, connected, locally Remarks: connected, separable or Lindelöf, then so is X/R. (i) If X is countable then X/R may not be countable. (ii) Separation axioms are not generally preserved. For example, the quotient space of a Hausdorff space need not be a Hausdorff space. 3.1.3 Let (X, ) be a topological space and let the quotient space (X/R, Q) be Hausdorff. Then R is a closed subset of the product space X X. 3.1.4 Let (X, ) be a topological space and let (X/R, Q) be the quotient space. If the quotient map Q is an open map and R is a closed subset of the product space X X, then (X/R, Q) is a Hausdorff space.
Chapter 3 22 Quotient Spaces of Topological Spaces 3.1.5 Let (X, ) be a topological space and let D be an upper semi-continuous decomposition of X and let D have the quotient topology Q. If G is a -open set containing a member D of D, then Q[G] is a Q-neighbor- hood of D. 3.1.6 Let (X, ) be a topological space and let D be a decomposition of X. Then D is upper semi-continuous if and only if the quotient map Q of X onto D is closed. 3.1.7 Let (X, ) be a topological space and let D be an upper semi-continuous decomposition of X such that every member of D is a compact subset of X and let D have the quotient topology Q. If X is (i) Hausdorff, (ii) regular, (iii) locally compact or (iv) satisfies the second axiom of countability, then D also has the corresponding properties. Let H be a subgroup of a group (G, .). Let the equivalence relation R in G be defined by xRy x -1 y H. Then given x G, the set { y G : yRx} is an equivalence class which we denote as xH = {xh : h H}; it is known as the left coset of H in G. Similarly, Hx is known as the right coset of H in G. When xH = Hx for all x G, H is said to be a normal subgroup. If H is a normal subgroup of G, the set of all cosets xH is a group known as the quotient group of G relative to H and is denoted by G/H.
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