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Question: uj is ilausaorff but not completely hausdorff b let k...

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uJ IS Ilausaorff but not completely Hausdorff (b) Let K- (x, y) E R: y 2 0, the upper half plane with the s-axis. Let Kı = {(x, 0) : x E R). i.e., the z-axis. Let K,-KV, For every (r,0) E K1 and r ER, r>0, let U((r,0),F) be the set of all points of K2 inside the circle of radius r centered at (z,0) and let U((x, 0) U(r,0), 1) u((x,0)) for i E N. For every (x, y) E K2 and r > 0, let U(x, y),r) be the set of all points of K inside the circle of radius r and centered at (r, y) and let U,((z, y)) U(x, y), 1) for i E N. i. Check the collection [B(r, y)r plex, where D((z,y)_ {U,((z,y)) : i E N. (a,y)E (BP1), (BP2), and (BP3). The unique topology on K is called Half-Disc Topology. K). has properties , ii. Prove that K is a T-space. iii. Prove that K is not a T3-space. Hint: Use theorem 6.12. ntudied

z C V, thus there exists k E N such that (1-1.1+1) it1ヒ Observe now that (-_-,ー+-) n( 2、1)メ0. Thus Vn( Therefore U nVメ0. ■ 51 C V J+2 j+1 1-1 6.11 Theorem: T3T2. PROOF: Exercise. 6.12 Theorem: A space X is regular if and only if for any r X and any open set V containing ar there exists an open set U such that EUCU cv. PROOF: Assume that X is regular. Let a be an arbitrary point in X Pick an arbitrary open neighborhood V of x. Then x X \ V where the later is closed. By regularity of X, there exist two open disjoint sets U and G with x E U and X \ V c G. Since Un G = ø, then Und-0. Thus Now, assume that the condition holds. Let A be any closed set and r be any point not in A. Then x E X \ A where the later is open. Thus by the condition, there is an open set U such that EUCUCX\A. Define V = X \ U. Then x E U, A-V and Un V-0. Thus X is regular. The space (X,T) is called a completely regular space if and only if for every T E X and every closed set F C X with a g there exists a continuous function f : X-1, where I = [0, 1] is considered with its 6.13 Definition: Let (X,T) be a topological space. Then

X , T) atlOn of t-Ure: X93r = {B : there exists an x E X with B E 93 t any neighborhood system B(r))rex must have all of the following properties: (BPI) For every x E X, B (x)メ0 and for every U ED(x), x EU.い (BP2) If x E U E B (y), then there exists a V E 93(a) such that V-U: (BP3) For any U1, U2 B(r), there exists a U, B(r) such that Us UinU 2.16 Example: In (R.u defina for

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