Show that all open balls are bounded subsets of a metric space.

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: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: are closed.

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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: Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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

A: Matric space

Q: Show that recel projeetrue space is an manifol a.

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Q: A metric space is sequentially compact if and only if it is totally bounded and complete. prove it

A: We need to prove that any given metric space is sequentially compact if and only if it is totally…

Q: What is norm space

A: In mathematics, a normed space or a normed vector space is a vector space over the real or complex…

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: 36) Prove that every product Space is inner metric Space- Discuss its Converse

A: Suppose that V is an inner product space over a field F. To prove that the V  is a metric space. To…

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: If S is a closed bounded subset of a metric space X, then S is compact.

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Q: Please define compact space and the clousure of .a set

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

A: A metric space X is compact if every open cover of X has a finite subcover.

Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Any cover of a metric space is open.

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

Q: Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete…

A: Show that closed ball Y in a metric space (X,d) is a closed set also show that if (X,d) is complete…

Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: Prove that any reflexive normed space is complete

A: A normed space X will be said to be complete if each Cauchy sequence in X is convergent in X.

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

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

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Q: Exercise. Prove that every finite subset of a metric space is closed set.

A: We have to Prove that every finite subset of a metric space is closed set.

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

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: 1. Show that any interval (a,b) in R with the discrete metric is locaaly compact but not compact

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

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

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

A: We have to prove that A finite set in a metric space can not have accumulation (Limit) point.

Q: Show that the closed ball Y in a metsic SPace (X,d)is a closed set. Also, show that it (x,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. also show that if (X,d) is…

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

A: Proof

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

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Q: Show that the metric. Closed ball in space closed a (X,d) Also, show Complete is Complete. a. set.…

A: To Show - Show that the closed ball Y metric space (X, d) is closed set. Also, show that if (X, d)…

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

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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 M, be set of all rare subsets of a metric space X. Prove that countable union of M, is rare.

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

Q: The intersection of any countable family of open dense-sets in a locally compact space X is dense.

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Q: 3: Prove that any reflexive normed space is complete.

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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.

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

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Q: (a) Show that every metric space is T2-space.

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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: Suppose metric space X is not path-wise connected then X is not connected. True False

A: A space is said to be Path connected if we can fine a continuous function between any two points of…

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.