Prove that the quotient of a locally connected topological space is locally connected.

Q: Show that every second countable (A2-space) space is separable.

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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: A topological space is Hausdorff iff limits of all nets in it are unique

A: This is a problem of Topology.

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: A topological space is Hausdorff iff limits of all nets in it are unique-

A: This is a problem of Topology.

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

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: 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: A ° = A – JA

A:

Q: 5. Prove that the property of a space having an isolated point is a topological property. 6. For any…

A: Sol

Q: Let X be a topological space

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Q: Define a compact space.

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

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: Every metric space is a hausdorff space .

A:

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: Prove that an affine map is continuous if and only if its linear part is continuous

A: If T:A→Rk is the affine map, where A is the affine span of an affine independent subset of Rn. Let…

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: Prove that linear maps are bounded.

A: To prove that linear maps are bounded.

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: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

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

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Q: Construct examples to show that if X and Y are topological spaces with G acting on them such that…

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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: Any cover of a metric space is open.

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

Q: If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this…

A: Given : (1) (ℝ, T) with the topology induced by taxicab metric.              (2) (ℝ, T) with the…

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

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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. 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: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: Show that any infinite subset A of a discrete topological space X is not compact.

A:

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

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: 1. prove every 2nd countable space is Lindelof space. 2. Every Lindelof topological space induced by…

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

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Q: a discrete space

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

Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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

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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: Finite topological spaces are always complete. True False

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

Q: Prove that a normed linear space is complete if its unit sphere is complete.

A: We prove that a normed linear space is complete if its unit sphere is complete. Its straightforward…

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

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

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Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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Q: Show that the closure of a connected space is connected.

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