Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable}. Then * None of the choices O (X,T) is not connected (X,T) is homeomorphic to (X,T1) where T1 is the finite closed topology on X O (X,T) is connected

Q: Let τ and τ' be two topologies of a set X. Is the family τ∪τ' formed by the open elements of both…

A: Union of two topologies is not necessary a topology.

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

A:

Q: 25. For any set S CR, let S denote the union of all the open sets contained in S. (a) Prove that S…

A: The given question is solved below.

Q: 2) Show that the digital line topology on Z is not Hausdorff.

A: Given that Digital line topology on ℤ, we have to prove that it not Hausdorff Therefore, Let a space…

Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

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: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

A:

Q: Find the general solution of the linear homogeneous constant coefficient differential equation if…

A: Just make sure you count all the roots.

Q: 5) We have seen that the collection of subsets of Z given by = {B(n) | B(n) = {n} when n is odd and…

A: I have just used the definition of basis and open sets in topology

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: possible

A:

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Let x be non-empty of points. a topological Space having open set with finite numb cal shase wh

A: Let X be a topological space having non-empty open set with finite number of points. Let Y be a…

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

A:

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: Theorem 2. Let R and S be binary relations on a set X. Then, (a) (R-1)-1 = R. (b) (RUS)-1 = R-'Us-!.…

A:

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

A:

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

A:

Q: 6) Let p and q be two points in a set, X. Consider the two collections of subsets of X: I. = {Ø, X,…

A:

Q: Exercise 3. Are the following subsets of the real line with its usual topology (in which U CR is…

A: