Determine whether or not the given set is (a) open,(b) connected , and (c) simply-connected. {(x,y) | 1≤ x2+y2 ≤4,y ≥0}

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: 4. Prove that if A and B are convex sets, then AN B is also convex.

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Q: Does there exist a non-empty open set A and two elements in A that are farthest apart (among all…

A: There are no two elements that are farthest apart in a non-empty open set. Justification : Let A…

Q: True or false? For all sets A and B,  (A∪B)'=A'∪B' This can be determined by constructing a…

A:

Q: 2. Sketch the following sets. Determine whether each set is open, closed, or neither and whether…

A: We will solve three subparts of the given problem as per the answering guidelines. If you want…

Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

A:

Q: Using contradiction method, prove that for every nonempty sets A and B, An (B- A)=0.

A: The given problem is to prove the following set identity using the contradiction method.

Q: Find an example of nonempty, distinct sets A, B, C for which (ANB) UC = (AUB) nC.

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Q: 21. Let A and B be two non-empty bounded sets of positive numbers and let C = {xy : x e A and y e…

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Q: А - (ВnС) %3D(А — В) n (А — С)

A:

Q: Every set S has an open cover.

A: If S is a subset of  R,   Then every set S  will always have an open cover.

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: - If C is Convex set ond a GR, then XC is a Convex set

A: Let C be a convex set and x∈R Then we have to show that xC is a convex set. The set xC is defined as…

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: Consider whether the following statements are true or false: a. The intersection of any two convex…

A: To state true or false

Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: 3. Using the definition of open sets, prove that an interval (0, a) is open where a > 0 is an…

A: (3) Definition of open sets: To define open set first we define an open ball For a real number x and…

Q: Prove that the unoin of a finite number of bounded sets is bounded.

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Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

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Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

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

A:

Q: A set consisting of single point has measure zero .A O nonzero measure .B O open .C O none of…

A: We have to find the measure

Q: Đ A is a closed set.

A: We prove by using the result that intersection of two closed set is closed.

Q: 6. Give an example of a set S which is (i) bounded above but not bounded below, (ii) bounded below…

A: (i). Let  S = { ..., -5, - 4, - 3, - 2, - 1 }

Q: Show that the interval [0,1] is a closed set in R

A:

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: let A Set bounded belou, erove that C. inf A Inf C A ン

A:

Q: let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are…

A: Given that X=a,b,c. To construct two topologies that have at least four open sets. We know that in…

Q: Find the cardinality of the set {x|x is an even whole number less than 10}.

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Q: Explain the idea of cardinality. When is a set denumerable? Show that the union of two denumerable…

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Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

A:

Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: 5. Differentiate with respect to x: SET A. 5e

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Q: Proof that the subset {(x, y)/x # 0, and y <-1} is an open set in R?-

A: Let S=x,y| x≠0, and y&lt;-1.  A subset S⊂ℝ2 is said to be an open subset, if its complement is…

Q: Show wheatear an open interval (m, n) is a union of a countable collection of closed sets or not.

A:

Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: b. Use the Sum Rule for two sets to show that |A UBUC| = |A| + \B| + \C| – |A U B| - |A U C| – |B U…

A:

Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

A:

Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Explanation of the answer is as follows

Q: 2- Moke the definitions of open and closed sets. x =Larbic,d,eļ and ノ closed subsets of (x,T) space,…

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Q: Show that if E is any measurable set, then -v (E) < v(E) < v*(E) and |v( E)| < \v|(E).

A: