. State whether the following is true or false: d) Every closed subset of a connected metric space is connected. e) The set R- {0} with the usual metric is connected. f) The set {3,0} with the usual metric is connected as union of connected sets {0} and {3}.

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.

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Q: Theorem 1.10. A set is infinite if and only if there is an injection from the set into a proper…

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

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

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

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Q: Show that the 4 incidence axioms that define a projective plane are independent.

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

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Q: thun no in

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

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

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Q: Let A be a subset of X, and B be a subset of X, where X is a metric space. Show the intersection of…

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