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: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

A:

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: Suppose A and B be subsets of metric space. Show that if A is compact and B is closed then A + B is…

A: Given A and B are subsets of a metric space. and also Given A is compact , and B is closed. We have…

Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

A:

Q: Show with all the relevant details, albeit succinctly that every compact metric space M is complete.

A: Here we use sequential definition of compactness so that we use that in proving the completeness of…

Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

A:

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

Q: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

A: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

Q: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

A:

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

A:

Q: Define metric space and define its four properties

A:

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

A:

Q: (a) Prove that every finite subset of a metric space is closed and has umulation points.

A: Since you have posted multiple questions, but according to guidelines we will solve the first…

Q: Let

A:

Q: Show that: All completions of a metric space are isometric.

A: According to the given information, it is required to show that all the completions of a metric…

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

A:

Q: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

A:

Q: Exercise. A finite set in a metric space cannot have accumulation points.

A:

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

A:

Q: 2.5 Show that any interval (a,b) inR with the discrete metric is locally compact but not compact.

A:

Q: The set [0,4] with the usual metric is sequentially compact. O True O False

A:

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: 1. Show that the real line is a metric space.

A:

Q: Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.

A:

Q: The set [0,4] with the usual metric is sequentially compact.

A: We have to decide whether the given statement is true or false.

Q: Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition…

A: 1. Let E and F be compact subsets in a metric space X,d. Let UE and UF be arbitrary open covers of E…

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

A:

Q: Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the…

A: Let X, d  be any metric space. It is required to prove that there exists a metric d such that d and…

Q: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

A:

Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

A:

Q: Prove that (R", II-|) is complete metric space.

A: This is a problem of Functional Analysis.

Q: Q7: Prove that (R", ||-|| ) is complete metric space.

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: ": Prove that ( R", ||||) is complete metric space.

A:

Q: Every closed subset of a connected metric space is connected. O True O False

A:

Q: Show that closed ball in metric space (x,d) is a closed set. Also,

A: Given: (x,d) is a metric space The objective is to show that the closed ball y in (x,d) is a closed…

Q: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

A:

Q: The set {0,3} with the usual metric is connected as union of connected sets {0} and {3} True False

A:

Q: Prove that in a metric space (X, d) every closed ball that is a set K(x, r) = {y e X : d(x, y) <=…

A: Question: Prove that in a metric space X, d every closed ball K¯x, r=y∈X, dx, y≤r is a closed set.…

Q: Prove that a normed linear space is complete if its unit sphere is complete.

A: We prove that a normed linear space is complete if its unit sphere is complete. Its straightforward…

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

A: Given that M is a metric space with metric d. Let Ba,ε be an ε-ball with center at a. That is:…

Q: Every closed subset of a connected metric space is connected. True False

A: