Say that X ⊆ R is closed under multiplication if for all x,y ∈ X, we have that xy ∈ X. Give three examples of finite subsets of R which are closed under multiplication. Give an example of an infinite proper subset of R which is closed under multiplication. Prove your answer. Give an example of an infinite proper subset of R which is not closed under multiplication. Prove your answer.

Q: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

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Q: 6. Show that the span of this set is all of R?. {}}

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Q: Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

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Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

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Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

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Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: or which compact sets A does there exist an m e N such that every open cover of A contains a bcover…

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

A:

Q: Prove that the complement of a dense open subset of R is nowhere dense in R

A: Prove that the complement of a dense open subset of R is nowhere dense in R

Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

A:

Q: In an indiscrete space X * O the only dense subset of X is X O every subset of X is dense in X a…

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Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

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Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: Show that if A is closed in X and B is closed in Y then A × B is closed in X × Y

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

A:

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

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Q: Prove that a set A is countable if and only if there exist a surjection from B onto A.

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Q: 2. Show that the set 3 S = 3 2 2 2 2 spans all of R³ or find an element of R' not belonging to…

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Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

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Q: compact, then E must be infinite. Exercise 6.46. Prove that if E is a closed subset of a compact set…

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Q: For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B…

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Q: A set L cR is convex if:

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Q: Prove that if Gn is an open set which is dense in the complete metric space (S,d) for n = 1,2,...…

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Q: Show that the following finite intersection property for a set A is equivalent to compactness: if B…

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Q: 1. Prove that a set SCR is everywhere dense if and only if S intersects every non-empty open set in…

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Q: Let A = {x ER|x' E Q} be a set. Is countable? Justify the answer by showing the explicit formula for…

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Q: Every proper simply connected oper equivalent to an open subset of the

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Q: (b) Prove that the complement of a dense open subset of RR is nowhere dense in R.R..

A: (b)  Prove that the complement of a dense open subset of R(R) is nowhere dense in R(R).

Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

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Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

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Q: Let A, B be disjoint infinite sets. Prove that their union A U B is denumerable if and only if both…

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Q: (b) Prove that the complement of a dense open subset of R is nowhere dense in R..

A: Prove that the complement of a dense open subset of R is nowhere dense in R.

Q: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

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Q: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

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Q: Prove that if A is closed and B is open, then A \ B is closed.

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Q: Show that every closed subset F of R' is the intersection of a countable col- lection of open sets.

A: detailed explanation mentioned below.

Q: For any collection a of subsets of X, there is ntaining a.

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Q: Prove that only ∅ and R are open subsets of R.

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Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

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