2.5 For each of the following properties, give an example consisting of two subsets X, Y CR', both considered as tope spaces with their Euclidean topologies, together with a map f : X →Y that has the indicated property.

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: 2) Find an example of a continuous map between two metric spaces which is not uniformly continuous.

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Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

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: 10. Suppose that U and V are finite dimensional vector spaces and that F:VU is a linear map. prove…

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Q: 1. In this exercise you will prove that disc {(x, y) : x+y < 1} is an open subset of R, and then…

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Q: get the following sets {(x, y) : x ∈[1, 2], y ∈[1, 2]} and {(x, y) : x = 2, y ∈[0, 1]}

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

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Q: Show that at a hyperboic poin, he principall direcions bisect the asymptotic directions using The…

A: Summary:At a hyperbolic point on a surface, the Gaussian curvature is negative, indicating that the…

Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

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Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

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Q: ) Give definition to the convex set of the space Rn. Also give a definition for when the function f…

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Q: For each of the sets find the set of boundary points and the set of interior points. Also state if…

A: We have, (a) [0,1]U{4} (b) x,y∈ℝ2:x2+y2=1 Here we have to find the  o find: Interior Boundary…

Q: Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

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