chapter 3 quotient spaces of topological spaces
chapter 3 quotient spaces of topological spaces
chapter 3 quotient spaces of topological spaces
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Chapter 3 22<br />
Quotient Spaces <strong>of</strong><br />
Topological Spaces<br />
3.1.5 Let (X, ) be a <strong>topological</strong> space and let D be an upper semi-continuous<br />
decomposition <strong>of</strong> X and let D have the <strong>quotient</strong> topology Q. If G is a<br />
-open set containing a member D <strong>of</strong> D, then Q[G] is a Q-neighbor-<br />
hood <strong>of</strong> D.<br />
3.1.6 Let (X, ) be a <strong>topological</strong> space and let D be a decomposition <strong>of</strong> X.<br />
Then D is upper semi-continuous if and only if the <strong>quotient</strong> map Q <strong>of</strong> X<br />
onto D is closed.<br />
3.1.7 Let (X, ) be a <strong>topological</strong> space and let D be an upper semi-continuous<br />
decomposition <strong>of</strong> X such that every member <strong>of</strong> D is a compact subset <strong>of</strong><br />
X and let D have the <strong>quotient</strong> topology Q. If X is (i) Hausdorff, (ii)<br />
regular, (iii) locally compact or (iv) satisfies the second axiom <strong>of</strong><br />
countability, then D also has the corresponding properties.<br />
Let H be a subgroup <strong>of</strong> a group (G, .). Let the equivalence relation R in G<br />
be defined by xRy x -1 y H. Then given x G, the set { y G : yRx} is an<br />
equivalence class which we denote as xH = {xh : h H}; it is known as the left<br />
coset <strong>of</strong> H in G. Similarly, Hx is known as the right coset <strong>of</strong> H in G. When xH =<br />
Hx for all x G, H is said to be a normal subgroup. If H is a normal<br />
subgroup <strong>of</strong> G, the set <strong>of</strong> all cosets xH is a group known as the <strong>quotient</strong> group<br />
<strong>of</strong> G relative to H and is denoted by G/H.