Let (Ka)aeA be a collection of compact subsets of a metric space, X. Show naeAKa is closed in X.

Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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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: are closed.

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

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

A: This is a problem of Topology.

Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

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Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

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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: Prove if {Galpha}alpha e A is a collection of closed sets in a metric space, then…

A: Recall that arbitrary union of open sets in X is open in X.

Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

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Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: 4. Let {Ka}aeA be a collection of compact subsets of a metric space, Show NaEA K, is closed

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

A: Topology Question

Q: in the discrete metric space: 1. All collections are clo

A: Let (X, d) be a metric space. Consider a closed ball Brx0 with both center x0 ∈ X and radius r >…

Q: 3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact

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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: Prove that the image of a continuous mapping of a compact set is compact.

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Q: Show that any compact set in R is closed and bounded.

A: Proof

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

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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: Let ||.|| and ||..|| be two norms defined on space X.Show that ||.|| and ||..|| are equivalent if…

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

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Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: 4- For any closed subset A of a metric space, A' C A 5- In a metric space, the intersection of any…

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Q: A discrete metric space X is separable if and only if X is countable

A: Given: To prove: A discrete metric space X is separable if and only if X is countable

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

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Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

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

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Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

Q: 1. Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: 2) Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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Q: Let X be a metric space. Let A and B be separated sets in X. Show either E is a subset of A or E is…

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