Let (X,T) be a topological space Property C=P.C.  A subset A of x has P.C If it's subset of the union of two disjoint nonempty open subsets of X then A is contained in only one of these open sets. Prove the following;  If A and B have P.C and A̅̅∩B≠ø  then A∪B has property c.

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

A: Let X be any infinite set with co-finite topology. Definition:  Let X be any infinite set. The…

Q: Suppose X and Y are subsets of some universal set. De Morgan's laws provide expressions for the…

A: X∪Yc=Xc∩Yc

Q: Exercise 2. Assume that (X, T) is a topological space. a) If A, B are subsets of X with AC B, show…

A:

Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

A:

Q: Linda planned to spend 9 hours practicing piano his week. By Tuesday, she had spent 2 1/2 hours…

A: Introduction: A fraction represents a portion of a total. This entire area can be considered a…

Q: Q. Let X = {a,b,c,d}, T = {o, X, {a}, {b,c,d}}. Then the topological space (X,T) is a. Normal and…

A:

Q: Given the following relation on sets : R={(x,y)1x is a proper subset of y}. Which properties hold…

A: The given relation:R=x,y| x is a proper subset of y.

Q: Let

A:

Q: 11. The complement of {0} in the Sierpinski space is A.open but not closed B.closed but not open…

A:

Q: A and B are open subsets of X with Cl(A) = X = Cl(B).

A:

Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

A: Use definition of disconnectedness mentioned below.

Q: Q. Let A = {(x, y), y = x, x E Z,y e R}, Then A is a. connected and pathwise connected space. b.…

A: d. neither connected nor pathwise connected space.

Q: (3) X is not the union of two disjoint non-empty separated sets. (4) X is not the union of two…

A: as given all the conditions are equivalent  (4) To show - X is not the union of two disjoint non…

Q: = {a, b, c, d}, t = {Ø, X, {b, c}}. Then in the topological space (X, t), {a} = %3D %3D %3D {a, b}…

A:

Q: give a Reel number r>0 and a∈ X in (X, ||.||) norm space. Determine whether B[a,r] is convex by…

A: Given that X is a normed space with norm . Let r>0, be a real number. Then closed ball Ba,r is…

Q: Consider the following sets. A = {0, 1}5 A₁ = {20 : x € {0, 1}4} A₂ = {201 : x ≤ {0,1}³} A3 = {2011…

A: A1 = {x0: x∈ {0,1}4}A2 = {x01 : x ∈ {0,1}3}A3 = {x011 : x ∈ {0,1}2}For the above sets to form a…

Q: Say whether (?, ?) is a topological space. Briefly explain your answer i. ? = R, and ? =…

A:

Q: 3. Let X be a topological space. Show that for every subset A of X and for ever U we have AQU =…

A:

Q: 5. Let X be a set and p E X. b) If 72 = {A C X : p ¢ A}U{X}is a topology on X, then determine…

A:

Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A: It is given that Xα:α∈J is an indexed family of connected topological space that is each component…

Q: Let (x,c) be a topologıcal space. Property C=P.C: A subset A of X has P.C if it's subset of the…

A:

Q: Q. Let S = {(x, y), x² + 4y2 = 4} CR2, then S - {point} is homeomorphic to – y? = 4} a. {(x, y), 4x2…

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

Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

A:

Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Prove or disprove whether or not the following two-dimensional set is convex.  S = { (x1,x2 ) such…

A:

Q: Let lx, T) be a topological Space and A, B S X- if A is open, then show that An CIIB) s CLLANB). if…

A: Given X, τ be a topological space and A, B⊆X. Suppose, A is open then An arbitrary x∈A∩clB implies…

Q: B) Let X = R be a metric space and consider the following subsets of X, %3D 1 1 S = (0,1), S2 %3D…

A:

Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

A:

Q: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: c) Let u be a usual topology on a set of real numbers R, and E= {2,3,6 Show that whether E is…

A: Given that R is endowed with the usual topology and that E={2,3,6}. We shall state a definition and…

Q: Let (X, d) be a metric space. a) Assume that A is a (finite or infinite) collection of open sets.…

A: To orove the required topological properties of open sets in a metric space (X,d)

Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

Q: 6. Prove that if x1, x2,..., Xn are distinct points in a Hausdorff space X. Prove that there are…

A: Let X be a Hausdorff space. Then in X any two distinct points are separated by disjoint open 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: State whether each of the following statements are true or false. If the statement is false, rewrite…

A: Part (i) If Then Hence So given statement is wrong . The correct statement is If Then

Q: disprove the following AUB s connected set then either A set prove on or B is conniedted

A: Since it is a multiple questions. As per our company guidelines, we have to solve the first one. For…

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

A:

Q: If V=R\power{3}, and W\index{1} is the xy plane and  let W\index{2} is yz-plane:…

A: Let (a,b,c) be any element of R3, then (a,b,c)=(x,y,0)+(0,y,z)=(x,2y,z) ⇒a=x, b=2y, c=z ⇒a,b,c∈R ⇒…

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.