4. Let A and B be two closed subsets of the connected topological space X. Suppose that X = AUB And that AnB is connected, prove that A and B are connected.

Q: We say that a space (X, τ) is not connected if there exist two nonempty openings A, B such that A ∩…

A: Open Set: A X, τ be a topological space. Then the sets in τ are called the open sets of the…

Q: Let X be any set with the discrete topology. Show that X is normal.

A:

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: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

A:

Q: 2. Show that totally disconnected is a topological property.

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

A: Topology Question

Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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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: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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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: Suppose X is a topological space having only a finite number of components. Prove that each…

A: given that  suppose X is a topological space having only a finite number of components . we have to…

Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

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Q: жат fuss vodo у льа изба и и & Dis PAS

A: To prove that an open ball in a metric space is an open set. Using the definition: Consider set A…

Q: Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is…

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Q: int topology (where p is the particular point). Let A= {b,p}. mpute the following: (i) the open sets…

A: Sol