Please define compact space and the clousure of .a set

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

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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: 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: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

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) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: Give an example of a set w-limit that is a minimal set.

A: To give an example of a set w-limit that is a minimal set.  A limit set is the state a dynamical…

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: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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

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Q: Define a compact space.

A: The objective is to give the definition of a compact space.

Q: 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: Define metric space and define its four properties

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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: (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: Prove the following: The countable product of metrizable spaces is a metrizable space.

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Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Any cover of a metric space is open.

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

Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

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Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: Hausdorff space.

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

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: The set S=[1,3) is not compact. Which open cover of S contains a finite subcover?

A: This is the question of Topology.

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: Show that an open subcover R is not compact by finding cover that does not have finite

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

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

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Q: Determine the dimension of the space spanned by the set

A: Given,               S=12-13, 1002, 28-48, 1111, 3306

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Remark: If (V,) is an inner product space. Then for all uiv,we Vand a,p EK い = &+p リ = す

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

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε>0 is…

Q: The closure of a compact subject of a regular space is compact.

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: Every proper simply connected oper equivalent to an open subset of the

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Q: If fis a bijection, find the sets X and Y.

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

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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: Show that the image of an open set under a continuous mapping need not be open

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

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Q: Give example closed set and bounded set is 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: Show that the closure of a connected space is connected.

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