Let (R,r) be topological space with usual topology on R, then the set Qof rationals is O Neither open nor closed Open O Closed Both open and closed

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: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: proof R with the finite-complement topology (F CT) is not continous to R with the usual topology.

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: A property Is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Given that, A property is said to be a topological property if it is preserved by homeomorphism:…

Q: A ° = A – JA

A:

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: 3) I Consider the topology = {GCR:0 ¢ Gor G = R} on R. Is open %3D %3D interval (-1, 1) compact in…

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Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that…

A: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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: let (x, m) be a measurable space. then ¥8:% [, o] are measural ble funetion, then ) I +9 is mea…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: Make the definition of tupological non-empty set and * E X. Inves tigale whether Space Let X be…

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

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Second option is correct

Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Topological property means it should preserve under homeomorphism,

Q: Let T be the cuclidean topology on R and r' be the finite-closed topology on R. Define f : (R, 7') →…

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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: {X, Ø, {1}, {2},{1,2},{1,3}}, is (X, t) a connected a) Let X = {1,2,3} with the topology t=…

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Q: 3) Consider the topology ↑ {GCR: 0 ¢ G or G = R} on R. Is open = %3D interval (-1, 1) compact in (R,…

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Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: ket (X, @) be a topokgical space, Ka compact subset of X and (FalneN a family in PCX) with Fr+ Ø POR…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: First option is correct

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that…

A:

Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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Q: In a Cofinite topology of X the collection of subsets is: a) r = {AcX|A±Ø or A is finite} b) r = {Ac…

A: Let X be a non empty set , then the collection of subsets of X whose compliments are finite along…

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: A property is said to be topological property if it is preserved by homeomorphism.

Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

A: As per our company guidelines we are supposed to answer only one asked question. Kindly repost…

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: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

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

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Here we need to identify why the boundedness and closedness are not topological properties.

Q: 5) Let 7 be the euclidean topology on R and 7' be the finite-closed topology on R. Define f : (R,…

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Answer :

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Given: A property is said to be a topological property if it is preserved by homeomorphism. Suppose…

Q: Show that the set of all real number constitutes incomplete metric space an we choose doxoy) = Itan'…

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Q: Let X be an infinite set with the countable closed topology T={S subset of X :X_S is countable}.…

A: We can show that X is countable. And by taking singleton we can solve the problem

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: First option is correct.

Q: Consider the subsets (0, 1) and [2, 3) of R, the set of all real numbers with the euclidean…

A: Consider the given sets. (0, 1) and [2, 3) We have to show that (0, 1) is not homeomorphic to [2,…

Q: Ris not connected if T is * O the indiscrete topology O None of the choices O the trivial (usual)…

A: Every indiscrete space is connected.  as the union of two disjoint non-empty sets, so R is a…

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/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent. 

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described. 

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.