Theorem 4.4.8. Let (S, d) be a metric space. A set A is open if and only if its complement R\A is closed. We can interpret this Theorem in the following way: complements of open sets are closed • complements of closed sets are open For a

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: are closed.

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: Assume E is disconnected. Then there exist two open set U and V in E such that 1. U∩V=∅ 2. E=U∪V 3.…

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: Suppose (X, d) is a nonempty metric space with a discrete topology. Show that X is connected if and…

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: Let A be a set and d : A × A → {0, 1, 2} a symmetric set function that satisfies d(x, y) = 0 ⇔ x =…

A: To prove that (A,d) is a metric space, we need to show that it satisfies the following properties:…

Q: 3) (i) Let X be a set containing elements a and b. Define T= {X, Ø, U| a E U and b E U}. This is a…

A: Definition:  A collection of subsets I of X is said to be a topology if it satisfies following…

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…

A:

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: 1. prove every 2nd countable space is Lindelof space. 2. Every Lindelof topological space induced by…

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: 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: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

A:

Q: 5. Give an example of a family {Un new} of open subsets of R, with usual metric, such that new Un is…

A: