State whether the following is true or false: a) The set [0,4] with the usual metric is sequentially compact b) The set [0, 1] with the discrete metric is compact. c) Any cover of a metric space is open.

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: A topological space is Hausdorff iff limits of all nets in it are unique

A: This is a problem of Topology.

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As…

A: Answer given below.

Q: Show that any totally bounded subset in a metric space is bounded.

A: Let A be a Totally Bounded Set Totally Bounded = For Each ∈ > 0, there exists an ∈-net. Let ∈>…

Q: are closed.

A:

Q: show that any infinite discrete topological space is not compact

A:

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.

A:

Q: State whether the following is true or false: g) The set of integers the usual metric is not…

A:

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: ) Prove the theorem that every metric space has a completion

A: Every metric has a completion.

Q: Given a metric space M with metric d, verify that any e-ball is an open set.

A: Given That : Metric space M with metric d . To Verify : Any ε-ball is open set.

Q: Prove that an infinite subset of a compact space has a limit point.

A: We can Prove an infinite subset of a compact space has a limit point.

Q: Q6) Find a topological space and a compact subset whose closure is not compact. Note hat a space is…

A:

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

A:

Q: B. Given a metric space M with metric d, verify that any ε-ball is an open set. (Please If possible…

A:

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As…

A:

Q: & and X are open sets.

A:

Q: rsection of any collection of com

A: This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the…

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: a)Every subset of a topological space is either open or closed. b)If X is a topological space with…

A: a)Every subset of a topological space is either open or closed.

Q: Exercise. Suppose that (X, p) is a metric space. a) Prove that any set in X consisting of precisely…

A:

Q: The set [0, 1] with the discrete metric is compact. O True O False

A:

Q: 5) Let X be a topological space. Is there always a metric on X which gives rise to its topology?…

A: No, there does not always exists a metric in a topological space that gives rise to its topology.…

Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

A:

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: Given a metric space M with metric d, verify that any e-ball is an open set.

A:

Q: thun no in

A:

Q: every finite subset of metric space is compact. ture or false?

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: State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. There are…

A: To Show: A set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.

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

A:

Q: . State whether the following is true or false: d) Every closed subset of a connected metric space…

A: