Define what it means for a topological space to be completely regular,

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

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Q: (a) Define a reflexive space. Prove that a Banach space is reflexive if and only if its dual space…

A: To Prove :-  (a) A Banach space is reflexive if and only if its dual space is reflexive. (b) Closed…

Q: Prove that continuous image of pathwise connected space is pathwise connected

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

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Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X

Q: Quertion:- Show that R is a Banach Space.

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Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…