Let (X, d) be a metric space and let y be a compact subset of X. Show that Y is closed and bounded.

Q: (3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and…

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

A: Sol

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: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: Q6) Find a topological space and a compact subset whose closure is not compact. Note hat a space is…

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Q: 4. Let XRx R, where R is the set of reals and let d be the Euclidean metric on X. Show that the set…

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Q: 2. Construct two metric spaces X and Y which are homeomorphic, but X is complete and Y is not…

A: 2. Construct two metric spaces  and  which are homeomorphic, but  is complete and  is not complete.

Q: One of the following statements is true:

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Q: The set [0, 1] with the discrete metric is compact. O True O False

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Q: 5) Let X be a topological space. Is there always a metric on X which gives rise to its topology?…

A: No, there does not always exists a metric in a topological space that gives rise to its topology.…

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: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

Q: (e) Show that a compact metric space can be covered by finitely many balls of any given radius. (f)…

A: A compact space has the property that every open cover has a finite subcover. In the context of…

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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Q: A of a metric space (x, d.)

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Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

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Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

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