Let X be an uncountable set and let oo EX be fixed. Let TX = {U E P(X); 0$U or X- U is finite} Let Y X-{∞} and denote by Ty the induced topology on Y, then (Y, Ty) is a compact space. True False

(2) Let (X,r)be a topological space, If for every A open set, and x E A, 3 Ux and Vy open sets, such that x E Ux, AC Vy, and Uxn Vy = 4. The space is called: (a) Hausdorff Space (b) Regular space (c) T1-space (d) Normal space

Exercise 3. Are the following subsets of the real line with its usual topology (in which U CR is open if and only if for every x ‹ U there is € > 0 with (x − €, x + € ) ≤ U) open, closed, both or neither? Explain. (1) [1,3); (2) (1,3) U (5,00); (3) (-∞0, ∞0); (4) { neN} U {0}; (5) The set Q of rationals. .

Theorem (Axioms of a Closure): Let (M,d) be a metric space and let S,T C M. Then (1) = Ø and M = M. (2) If S CT, then ŠST. (3) 5 is the smallest closed set in M such that S S. (4) S is closed in M S = S. (5) Š = S. (6) SUT = SUT. (7) In general, (Sn T)snT. But, (Sn T) 5nT. (8) S = 5e.