Is it possible to have a function on a metric space which is discontimuous at every point of the metric space but the restriction of that function on a dense set is continuous? Explain.

Q: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: Remark. For everyr E N, a set in R" with the Euclidean metric is com- pact if it is bounded and…

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Q: What is the continuous extension?

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

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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: is continuous at (0,0). 3. Show that X = [0, 1] with the discrete metric is bounded but not totally…

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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: Show that a uniformly continuous function on (a, b) is bounded.

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

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Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: space is the union of a finite number of closed balls radius e. Prove that a metric space is totally…

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Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded. %3D

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Q: Let X be a limit point compact metrizable space. (a) Show that there exists no continuous,…

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Q: 3. Show that X = [0; 1] with the discrete metric is bounded but not totally bounded

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

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Q: Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: Verify that the discrete metric is actually a metric.

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Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: give example is not To-SPace about to folegical SPace

A: We have to find a topological space which is not t0.

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: 3) Consider the topology ↑ {GCR: 0 ¢ G or G = R} on R. Is open = %3D interval (-1, 1) compact in (R,…

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Q: Show that the inner product (,) on an inner product space X is a bounded sesquilinear form h. What…

A: Given that X is an inner product space with an inner product ·,· defined on it. We know that a map…

Q: 2.5 Show that any interval (a,b) inR with the discrete metric is locally compact but not compact.

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Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

Q: A bounded function f which has only a finite number of points of discontinuity in [a,b] is…

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Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

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Q: 1. Show that any interval (a,b) in R with the discrete metric is locaaly compact but not compact

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

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

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Q: What is the Boundedness Theorem?

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Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: a) Show that, any function f: X → Y is continuous if X have discrete topology.

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

Q: Show that closed ball in metric space (x,d) is a closed set. Also,

A: Given: (x,d) is a metric space The objective is to show that the closed ball y in (x,d) is a closed…

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: 3. Show that X = [0, 1] with the discrete metric is bounded but not totally bounded.

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Q: Find the bounded of the determine red In the same way as the proof of theorem 4 in the second…

A: Given:                                            Sn+1=Sn-paSn-q+bSn-r+cSn-sdSn-q+eSn-r+fSn-s…

Q: a function f is continuous at x0 element of X if and only if f is both lower semi continuous and…

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Q: nach limits to show that the space ℓ∞ is not reflexive.

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Q: Consider the topology 7 = {GCR:0 ¢G or G = R} on R. Is open interval (-1, 1) compact in (R, T)?…

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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: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: Let X and Y be normed linear spaces and F : X → Y be linear. Prove that F is continuous if and only…

A: Let X and Y be matric space and let f: x→y be a uniformly continuous function. If ( xn)n∈N is a…

Q: What is the continuous extension of

A: This question is taken from the function and derivative now we have to solve the given function so.…

Q: Show that the space C(R) of all continuous functions defined on the real line is an…

A: According to the given information, it is required to show that C(R) of all continuous functions…

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

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Q: Let (X, d) be a metric space and let f, g : X → R be two continuous functions. (a) Prove that the…

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