If T1 and T2 are two topologies on the same space X, then * T1NT2 and T1UT2 are both topologies on X T1UT2 is a topology on X but T1NT2 is not TINT2 is a topology on X but T1UT2 is not T1NT2 and T1UT2 both cannot be topologies on X

Q: Let's say that a collection of open balls (Ui)ie1 = (Bx, (ri))iel in Euclidean space Rd is = UB₂,…

A:

Q: . Is it true that the null set is a subset of every other set? (Y/N)

A: If a set has no element then the set is called null set. Null set is also called empty set. It is…

Q: 4. Let A and B be two closed subsets of the connected topological space X. Suppose that X = AUB And…

A: Method of contradiction : suppose A is disconnected ,then there are two disjoint closed non empty…

Q: Show that compact sets are closed under arbitrary intersections and finite unions. (Here we're…

A: To show that compact sets are closed under arbitrary intersections and finite unions

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

A: We will define a bijection from the set J  = [ 0 , 12 ) ∪(12 , 1]   to the set I =0 ,1   First we…

Q: We say that a space (X,τ) is connected if there exist two open non-empty spaces A,B such that A∩B≠∅…

A: Let X,τ be a connected metric space. We have to show that the only subset of X are both open and…

Q: 10. Prove that for subsets X, Y, and Z of a given universal set U (a) X-Y=X-(XnY)= (XUY) - Y (b)…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: To give an example of subsets A,B in a topological space X such that both A,B are connected but A∪B…

Q: Show that any finite union of Nowhere Dense Sets is Nowhere Dense.

A:

Q: 5.6.34. (a) Find an example of set A and a nonempty subset B C A, such that A and A \ B have the…

A:

Q: 8.(a) Give a direct proof that when w, x are real numbers such that w|sx|, we have \w + x] 2 (b) s…

A:

Q: Show that the only Hausdorff topology on a finite set is the discrete topol- ogy.

A: The question asks us to prove that the only possible Hausdorff topology on a finite set is the…

Q: Consider the set of lattice paths from (0, 0) to (8, 8). You should know one quick formula for the…

A: Using formula,nk=n!n-k! k!i 80=8!8-0! 0!=8!8!=1ii 71=7!7-1! 1!=7×6!6!=7iii 62=6!6-2! 2!=6×5×4!4!…

Q: For each of the following sets, decide whether it is finite, countable, or uncountable. Explain your…

A: Let  be any set , then

Q: possible

A:

Q: (a). Show that (0, 1) × (0, 1) and (0, 1) have the same cardinality. (Hint: shuffling two digit…

A:

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

A:

Q: Can a topology on an infinite set not be discrete and indiscrete in which every subset is clopen?

A:

Q: Are the following true or false? If true, prove. If false, give a counterexample. (a) The…

A:

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: Disconnected sets

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: (a) Represent the set {x = (x1, x2) = R² | x1x2 ≥ 1}, as the intersection of some family of…

A: The given set is x=(x1,x2)∈R+2|x1x2≥1. To Find: a) Represent the set x=(x1,x2)∈R+2|x1x2≥1 as the…

Q: How can I show that the collection  A composed of empty set and all cofinite subsets of X Is a  sets…

A: