6. Every finite set in R is compact. Why?

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: Do the sets [0, 1] and [0, 1/2) ∪ (1/2, 1] have the same cardinality? Either give an explicit…

A:

Q: Let A be a set and let B be a proper subset of A. Is it possible for B to have the same cardinality…

A: Given: Let A be a set.And B be a proper subset of A.

Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: 7. Show that if S x T is a closed subset of R?, then S and T are closed subsets of R.

A:

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

A:

Q: 1. Let A, B and C be the subsets of U. Then prove that (A – B) - (B - C) = A-B by using laws of sets

A: Using properties of set theory

Q: 4. Let S be any nonempty set with the operation a * b = a. Is (S,*) a semigroup? Is it commutative?

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: 8. Disprove that every compact set has only a finite number of isolated points. V Note that the…

A: Let C be the compact set.If possible suppose that 'C' has a infinitely many isolated points. say…

Q: 2. Show that Z has the same cardinality as the set O of odd integers.

A: Given below the detailed solution

Q: 1) Define a subset.

A: To define a subset.

Q: 3.1 Show that a finite intersection of closed sets is closed.

A:

Q: A set S is denmberable if S does not have the same cardinality as P. ylgn ihi

A: A set S is denumerable if S does not have the same cardinality as P

Q: d. What is the cardinality of a countably infinite set?

A:

Q: Suppose that A and B are sets. A is a subset of X and B is also a subset of X. Please teach me how…

A:

Q: 2- Prove that H, is convex set.

A: Let us look at an arbitrary closed half-space defined as H+={x∈ℝn:aTx-b≥0}, representing a half…

Q: 2. Show that any subset of a countable set is countable. 3. Show that a nonempty set S is countable…

A: Hence any subset of a countable set is countable.

Q: Suppose that the universal set U is the set of all even intergers from 1 to 16 inclusively and A =…

A: First we will discuss the properties of a Complement set. Complement of a set X is denoted by Xc or…

Q: Can a set and it's Complement be both nowhere dense in R? A Ac=R-A

A:

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.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

Q: 4,P) is a aɔnds (a) The intersection of any collection of closed sets in X is closed (in X). (b) The…

A:

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

A: Que  Ans :  Given ( X, ||-||) be a norm linear space,  A be any open subset of X    ( def- set A is…

Q: 8. (a) Show that the intersection of finitely many open sets is open.

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Suppose A, B, C are sets. what do you need to show to prove the fllwing definition of subset:…

A: If x is a general element in P ,the given statement implies that each element in P is also in Q…

Q: If a finite set A equals the union of k subsets A1, A2,..., Ak, then the number of elements in A…

A:

Q: 10. The supremum of a non-empty set S, if it exists, is always unique.

A: We need to prove that Supremum of a non empty set S, if it exists, is always unique.

Q: 1. Prove that, if A, B, and C are sets and if ACB and ACC, then ACBOC.

A: To prove: A⊆B∩C

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: 5.3.8 Prove that if A CR is a finite set, then A is well-ordered.

A:

Q: 15. We have seen that an infinite set can be put into a one-to-one correspondence with a proper…

A:

Q: Let {S,}EN is a disjoint seqeuence of measurable sets , then: US, µ(S,)

A: Given a statement

Q: Prove that Everry subset of negigible seti's gigible

A:

Q: 20. What is the cardinality of each of these sets? а) и c) {Ø, {Ø}} b) {Ø} d) {Ø, {Ø}, {Ø, {Ø}}}

A:

Q: O No 4- Let x bea and That sad anly Aeplegical Space sibspace fxiShow chred a subset' C of y is if…

A: Y be a subspace of X means that any open set in Y is the intersection open set in X with Y

Q: (a) The intersection of an arbitrary collection of closed sets is closed. (b) The union of any…

A: (a) We have to prove that the intersection of an arbitrary collection of closed sets is closed. Let…

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

A:

Q: If a set B has 64 proper subsets of odd cardinality, what is |B|? *

A: Total no os subsets of any set A, with cadinality 'n' is- 2^n Now the no of proper subsets of of A,…

Q: 5. Show that a finite set is a closed set.

A:

Q: prove that the union of a countable set and an uncountable set is uncountable

A:

Q: Do the sets [0,1] and [0,1/2)∪(1/2,1] have the same cardinality? Either give an explicit bijection…

A: Yes, the sets 0,1 and 0,12∪12,1 have the same cardinality.

Q: is the set [5,100] compact?

A: Given that:A set is S=5,100.We have to check whether a set 5,100 is compact or not?

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: Show that if A is a countable set, and B CA is infinite, then B is countable. [Hint: How could we…

A: To prove that  If A is a countable set and B is a subset or equals to A is infinite then B is…

Q: Prove that if A1 and A2 are compact sets then A1 U A2 is compact.

A: Given that A1,A2 is compact sets. The objective is to show that A1∪A2 is compact. Now let's take an…

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: 2. Prove that the union of any finite collection of closed sets in RP is a closed set in RP.

A: A set A is closed if and only if every limit point of A belongs to A. Now we can prove the result.…

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

A:

Q: prove that every subset of negligible. set is negligible.

A: I have used definition of negligible set.

Q: 6. Suppose that U is an infinite universal set, and A and B are infinite subsets of U. Answer the…

A: U is an infinite universal setA ⊆U is infiniteB ⊆U is infinite