3. (a) Let (R, τe) be the usual topology on R. Find the limit point set...
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set of the following subsets of R (i) A = { n+1 n : n ∈ N} (ii) B = (0, 1] (iii) C = {x : x ∈ (0, 1), x is a rational number
(b) Let X denote the indiscrete topology. Find the limit point set A 0 of any subset A of X.
(c) Prove that a subset D of X is dense in X if and only if int(X\D) = ∅.
(d) Prove that a space X is compact if and only if it has a basis β for which every cover of X by members of β has a finite subcover.
Homework Answers
![4 = {+ point of A is a EN] A is discrete. So, no limit point. B = (0,1] (tu The set of limit points of B = c/B) = [0,1] - (](http://img.homeworklib.com/questions/fb54f0e0-d5ed-11ea-bf8b-0f6a8145e7be.png?x-oss-process=image/resize,w_560)
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3. (a) Let (R, τe) be the usual topology on R. Find the limit
point set...
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usual discrete indiscrete finite complement Lower limit countable complement K-topology 0 0 connected path connected O compact countably compact limit point compact B-W compact sequential compact 1st countable 2nd countable LindelÖf 0 Ti T2 separable
usual discrete indiscrete finite complement Lower limit countable complement K-topology 0 0 connected path connected O compact countably compact limit point compact B-W compact sequential compact 1st countable 2nd countable LindelÖf 0 Ti T2 separable
5- Recall that a set KCR is said to be compact if every open cover for...
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Real Math Analysis Let A be a nonempty finite subset of R. Prove that A is compact. Follow the comment and be serious Please. our goal is to show that we can find a finite subcover in A. However, I got stuck in finding the subcover. It is becasue finite subset means the set is bounded but it doesn't mean it is closed.
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Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
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3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the con...
Let X be a set and let T be the family of subsets U of X such
that X\U (the complement of U) is at most countable, together with
the empty set. a) Prove that T is a topology for X. b) Describe the
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Topology (b) Let S denote the subset of co consisting of sequences with rational entries of which at most finitely many are nonzero. (i) Show that S is dense in co with the sup norm. [Hint: Show that...
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(b) Let S denote the subset of co consisting of sequences with rational entries of which at most finitely many are nonzero. (i) Show that S is dense in co with the sup norm. [Hint: Show that for every r E co and every ε > 0, there exists y S such that llx-yI100 < ε.j (ii) Conclude that (co, ll . 114) is separable (only quote relevant results) (iii) Show that the closed unit ball in (a-II ·...
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Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
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