let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are equivalent (and show why they are)

Q: Prove that a subset of a set of measure zero has measure zero.

A:

Q: If  X is a countable set show that every topology on X is second countable.

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

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 ,…

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

A: No, there is no set A that satisfies A={A}. Let A is an empty set, i.e.  A=  or  A=ϕ, in which…

Q: Let A be the set of even positive integers less than 40 and divisible by 4. What is the cardinality?

A: This is the question from set theory.

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

A:

Q: Suppose that S is a non-empty finite set. Prove that ? has as many even-sized subsets as it does…

A: Let S be a non-empty set. Let A be the set of even sized subsets of S and let B be the set of odd…

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: Show by membership that for all sets A, B and C: A-(ANB) CA-B

A: Sets and relations

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

A: Suppose a set M is closed and bounded. Then we have to show that M has a maximal and minimal…

Q: Assume that the set S has 14 elements. How many subsets of S have at most 4 elements

A: At most 4 means 4 or less.  Let us denote X=number of elements

Q: Show

A: We prove this statement by using definition of Borel Set.

Q: Let A {a1, a2, ...} and B = {b1,b2, ...} be countably infinite sets. %3D Prove that A × B is a…

A:

Q: Let A and B be two subsets of E. Prove using the set identities: А - В — В — А.

A: You have asked multiple unrelated questions in the same post. I have addressed the first one. Please…

Q: Use the element method for proving that a set equals the empty set to prove that for all sets A, B…

A:

Q: Prove that empty set is subset of all sets: φ ⊆ A

A: Given: An empty set and set A is given. To Find: To prove that empty set is subset of all sets: ϕ⊆A

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

A:

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: Show that if A is an infinite set, then whenever B is a set, AUB is also an infinite set.

A: we have to prove that AUB is infinite set.  Since, given A is infinite set. since we know that…

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: Suppose that X and Y are sets such that the power set of Y is a subset of the power set of X. Does…

A:

Q: Assume that the set S has 13 elements. How many subsets of S have at most 4 elements?

A:

Q: Prove that A is an infinite set if and only if ∀n ∈ ω, A has a subset B such that B ≈ n.

A: given that A is an infinite set let B be any set let prove this contradiction suppose assume that B…

Q: Prove that if A is a set then so is {A}!

A:

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

A:

Q: If A is set of all vowels in alphabets.Then number of subsets of A is

A: Given: A contains all vowels  So, A = a, e, i , o , u Number of elements in A = 5

Q: Prove that every infinite set S contains a proper subset similar to S.

A: We have to prove that every infinite set S contains a proper subset similar to S. Let X=S Let X be…

Q: Use the definition of set operations to prove that for any set A, A- A = Ø . A + A = Ø

A: To prove: Using definition of set operations, for any set A prove that                  a) A-A=ϕ…

Q: Consider all subsets of size four of the set {1, 2, ..., 20}. How many of them have a nonempty…

A: Given: It is given that by considering that all the subsets of size four of the set{1,2,....20} The…

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

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

A: Set equality

Q: Consider the following theorem of set theory: For any two sets A, B, [A U (B – A)] = A U B. a.…

A:

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

A: Here is the proof .

Q: The closare of a set is smallest closed set containing s صواب ihi

A: Let S be a subset of X: S⊆X. Let M be the closure of S. Then M is the smallest closed set which…

Q: open

A:

Q: Prove that if A is a set then so is {A} but do NOT use an argument that involves stages explicitly.

A: 3 Given A is any set. We have to prove that A is also a set. Since, A is any set Then as every set…

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Construct a family of sets that demonstrates R is not compact

A:

Q: Let A be a well-ordered set, then every subset of A is not well-ordered

A:

Q: 4. Prove that the set Z is a closed set in R.

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: Prove that every set with positive Lebesgue measure contain a non-measurable set

A:

Q: The cardinality of a finite set A, denoted by Al, can provide the of A as equal to 21A

A:

Q: Let S be a set of n elements. Find the number of distinct subsets of S.

A: Given: Let S be a set of n elements.

Q: Prove that the set of all finite subsets of ω is countable. (Set theory)

A: The objective is to prove that the set of all finite subsets of ω is countable.

Q: Show that Z has the same cardinality as the set E of even integers.

A: In order to prove that ℤ and set of even integers E have the same cardinality. (i.e) we need to find…

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…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…

Q: Let A and B be sets. Define precisely (but concisely) what it means for A to be a subset of B

A: Let, A and B be setswhat it means for A to be a subset of B.