True False An open interval is not homeomorphic to either a closed interval or a half open interval. * True False A complete metric space is connected. True False

Q: Give an example of an infinite collection of nested open sets in RR under the usual metric on RR…

A: We given an example of infinite collection of nested open sets in R such that their intersection is…

Q: 3.2 Find all limit points of a set B = {1, },, 1, ...} on the real line with the usual metric. Is B…

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Q: الموضوع : theorem: Let fix_y applied between two topological spaces (X,Tx) and (Y, Tj), then the…

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Q: 0:1 What are the basic properties of finite dimensional space, explain that briefly.

A: We are supposed to answer only one question.

Q: 3.2 Find all limit points of a set B = {1,,, i,...} on the real line with the usual metric. Is B…

A: The objective is to find all the limit points of the Set B and to check whether B is open or closed.…

Q: Exercise 33 (i) Let A = (a, b) be an arbitrary open interval. Write A as a countable union or a…

A: Set based ques.

Q: The set of integers the usual metric is not compact. True False

A: In usual metric a set is compact iff it is closed and bounded

Q: Let X be a locally compact Hausdorff space which is not compact; Let X* be one-point compactificatin…

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Q: If X= R and d be the discrete metric Then the set of natural number (N, d) is complete no decision…

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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: Prove there exists a continuous surjection from the unit interval to the 2-sphere

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

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Q: ) Prove the theorem that every metric space has a completion

A: Every metric has a completion.

Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: Lemma 2.5.4 Let (X, d) be a metric space. Then Both 0 and X are open in X.

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

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

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Q: a)Every subset of a topological space is either open or closed. b)If X is a topological space with…

A: a)Every subset of a topological space is either open or closed.

Q: Exercise. Suppose that (X, p) is a metric space. a) Prove that any set in X consisting of precisely…

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Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

Q: w1 is normal prove it prove that the first uncountable ordinal omega 1 is normal space by showing…

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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: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

Q: M is an infinite set. If x,y ∈ M, explain the relation f below            {  1 , if x = y   f(x,y)…

A: We have to solve given problem:

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

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Q: Assume that A is a compact subset of a metric space (X,d), and B is a closed subset of A. Line 1:…

A: Combined all line with last line and check which makes a meaningful true statement

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

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Q: Is it possible to have a function on a metric space which is discontimuous at every point of the…

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Q: Lemma 2.5.6 Let (X, d) be a metric space. Then 1. Both and X are closed in X. 2 Any intersection of…

A: Given that, (X,d) be a matric space.  A space is a non empty set together with a metric on the set.…

Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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Q: Prove or give counter example an infinite product of discrete spaces may not be discrete

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Q: ii) True or False: The space R is homeomorphic to the closed interval [-1, 1] with the usual…

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Q: Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

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 ordered pair is an example of a topological space that is connected but not path-connected,why?

A: We use contradiction method.

Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

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Q: Set of Space. d f Real number R is veeter An d is Spale

A: We know that ℝ,+ is an abelian group. Now define scalar multiplication on ℝ by α.a=αa where…

Q: Show that a union of any finite points of bounded subsets of a metric space is bounded

A: One way to prove this result is to take each set in the union and pick a single point to be its…

Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

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

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

A: Yes, any discrete space is complete.

Q: 3. Consider the metric space (R..|) and let S be the union given by (0}U UI J. Note that each of the…

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Q: Let K and L be two compact subsets of a metric space, then: O KUL and KNL are both compact. O KUL is…

A: (i) Any open cover of X1 ∪ X2 is an open cover for X1 and for X2. Therefore there is a finite…

Q: Prove that every normed space is continuous and moreover uniformly continuous function. Give an…

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