Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not ?

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A: To Show: If  X is a countable set show that every topology on X is second countable.

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A: See the attachment

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

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Q: Show that every open set with respect to 7 is a union of open ball(s).

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Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: Prove that if S is a bounded set then S is bounded.

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Q: Show that the o-algebra B of Borel sets contains all F,-sets.

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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: be an open subset in C and f

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Q: 4. Let A be an open set in å nörmed space subset B of X .

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

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

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Q: Write up a formal proof of the following: Claim: For any set A, A ⊆ A.

A: To prove rigorously that any set A is a subset of itself.

Q: Prove that the Cantor set has measure zero.

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Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set.

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Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not exactly

A: Set equality

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Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

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Q: Show that the derived set E′ of any set E is closed.

A: Here is the proof .

Q: A set L cR is convex if:

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Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: open

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Q: )Š is the smallest closed set in M such that S CS.

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Q: Prove that if A is a transitive set then UA is a transitive set.

A: If A is a transitive set then, It is proved that ∪A is a transitive set as shown below.

Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

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