P is regular if all Borel sets are regular. Define C to be the collection of regular sets. (a) Show R e C, Ø e C. (b) Show C is closed under complements and countable unions. (c) Let F(R) be the closed subsets of Rk. Show

Q: Theorem 8.3. The space Rstd is connected.

A: To prove: The space ℝstd is connected. Proof: On the contrary assume that ℝstd is not connected that…

Q: 4. Let X be a set. Let A, B be the elements of the power set of X, P(A). Define the symmetric…

A: When we collect one or more well defined object and keep them together, it forms a set. The theory…

Q: Theorem 1.10. A set is infinite if and only if there is an injection from the set into a proper…

A:

Q: Argue that a set A is a Gδ set if and only if its complement isan Fσ set.

A: The given statement is of Baire’s Theorem.

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: How do I write a set T such that S is an element of T and a proper subset of T?

A:

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Show

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

Q: A closed set in the finite complement topology on R.

A: Def: A set C is closed in topological space, if complement of C is open set.

Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

A:

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: For each of the following collections of sets, define a set An for each n N such that the indexed…

A: (a) From given collection of sets,we can say ,In other words, given collection is nothing but now…

Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: Let A and B be open sets in a topological space. Is A ∩ B open?

A:

Q: Prove T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set

A: We have to prove the following -   T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set.

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

A:

Q: Let A, B, C be non-empty subsets of the universal set U such that (An B) CC. Prove that AC BUC.

A: Let x∈A, then either x∈B or x∉B.Case 1: If x∈B ⇒x∈B∪C.⇒A⊆B∪C.