Please define these concepts: normal space, .Hausdorff space

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

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Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: Find an open cover of the interval (−∞, 3]. Justify why that open cover has no finite subcover.

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Q: Let (R. d) be discrete metric space, then Ris separ able O True O False

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Q: what is reflexive Banach space ?

A: Banach space: A normed linear space N,· is called banach space if it is complete metric space. A…

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

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Q: Please define these cocepts: Isolated point, .discrete topology, dense space

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Q: What is the difference between usual metric and unusual metric space?

A: Define any other metric then usual one.

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 (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: the quotient space of a T0−space necessarily T0

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Q: Let X be a topological space

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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: Define the term dimension of a space.

A: In mathematics, the dimension of space means the minimum number of coordinated to specify or located…

Q: Find an example of a topological space that is seperable and first countable but not second…

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Q: State true or false with brief justification: L1[a,b] is a reflexive space

A: We use the Theorem: Let X be a separable normed space. If the dual space X* of X is non-separable,…

Q: Verify that the discrete metric is actually a metric.

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Q: Exercise. Every point in a discrete metric space forms an open ball.

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Q: Restate the contraction mapping principle, and list what it tells us about the mapping g.

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Q: What is the norm of a partition of a closed interval?

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Every inner product space is metric space and normed space O metric space but may be not normed O…

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Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

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

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Q: Who was the first to make use of an abacus

A: `Who was the first to make use of an abacus?`

Q: Every point in a discrete metric space forms an open ball.

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Q: Let [a, b] be an inter a space. Let ƒ:[a,b] → X

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Q: The usuall metric space (Rd)isComplete ***

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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: Describe the largest set S on which it is correct to say that fis continuous. AX, y) = In(1 + x² +…

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Q: What is Euclidean Space??

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Q: Choose two figures which are topologically equivalent to each other 20 L.

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Q: ovide an illustration of the statement "Every normal space need not be a ular space."

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Q: find the areas of the regions enclosed by

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Q: What is the Boundedness Theorem?

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Q: What is Sierpinski Space? Also give example

A: Sierpinski Space is  a topological space. Now to define a topological space we have to define a…

Q: a discrete space

A: We know that in a discrete topological space X, every point in X is an open set.

Q: What is the size of the space stretched by the set {x, x+x², x³, 1, x²}? a) 1 B) 4 C) 0 D) 34…

A: size of the space streatched   by the set of n element is 2n+2

Q: be a n-dimensional normed space, then prove that its dual space is also n-dimensional.

A: Let X be a normed linear space. Let X* be its dual space with the usual dual norm ||T|| = sup{…

Q: Task 13 Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: Every compact, Hausdorff space is normal.

A: To prove: Every compact, Hausdorff space is normal. Let X be a compact Hausdorff space. Let A,B⊂X be…

Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f…

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Q: Prove the following: Let (Y, d) be a metric space. Let B be a bounded subset of Y. If A C B, then A…

A: Let Y, d be a metric space and B be a bounded subset of Y. Suppose A⊂B. To prove that A is bounded…

Q: What is Null Space?

A: We will solve the given question by giving the definition of null space. Let U and V be two vector…

Q: Let X and Y be two discrete spaces, then *

A: Note: Since you have posted multiple questions we are only allowed to solve the first one. Thankyou…