Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A) (resp. iso(A)) denote the set of isolated points of A (resp. A). Is it true that iso(A) = iso(A)?

Q: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

A:

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

A:

Q: Prove that Q is not isomorphic to Z.

A:

Q: Let Y be an infinite subset of a compact set X CR. Prove Y' 0.

A:

Q: sketch the set {z ∈ C : Re(z) = Im(z + 2 + i)}

A: We have to sketch the set {z∈ℂ:Re(z)=Im(z+2+i)} Let z=x+iy where x,y∈ℝ

Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

A:

Q: Let Y be an infinite subset of a compact set XC R. Prove Y' 0.

A:

Q: (a) State the definition of an antichain A of a set S.

A: As per guidelines only first question is solved. For remaining question kindly repost the question…

Q: 1 Properties of the Cross Product: lamat, 1.) ||lox b|| = |lo||-·||bl| - sind, where O & [alt] is…

A: As this question has multiple subparts, hence only first five subparts will be done.

Q: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

A: Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this…

Q: Let Si = { x ∈ R ∣ x > i } , i=0,1,2,... Are the sets S1, S2, S3 . . .  are mutually disjoint?

A: If two sets A and B are mutually disjoint, then A∩B=ϕ . Define the sets S1 , S2 , S3 as follows.…

Q: Suppose ƒ(.) : R" → R is proper. Show that f(-) is convex i and only if epi(f) is a convex subset in…

A: