10. Let X and Y be topological spaces. Show that if either X or Y is con- tractible, then every continuous map from X to Y is homotopic to a constant map.

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

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Q: Let X and Y be topological spaces and fix some yo E Y. Show that the map f: X→XXY defined by f(x) =…

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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: Let f: A --> B and g: B --> C be maps. Show that if g of f is 1-1 and onto, then f is 1-1 and g is…

A: According to the given information,

Q: Definition. Let X be a topological space and suppose xo is a point in X. A map exo : [0, 1] → X that…

A: To prove the theorem,  First define the homotopy function  H:I×I→X by Hs,t=α2st+1    0≤s≤t+12y…

Q: Let f: X→ Y be a map between the sets X and Y. (a) Show that f is injective if and only if there…

A: (a). Let’s first assume that F is an injective map.We need to show that there exists a map  such…

Q: Let X, Y be the non-zero normed spaces. Then for any non-zero element x = X, there is a bounded…

A: Let's construct the linear map T:X→Y described as follows.Since x is a non-zero element of X, let's…

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: Exercise 11.2. Prove Proposition (b) An injective covering map is a homeomorphism.

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Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, 𝛕X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: Write a pair of curves that are homotopic and show that they are

A: it is known that two curves are said to be homotopic if one is deformed into another continuously.…

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

A: We have to show for every closed subset C , f(C) is closed in X×X

Q: ► Exercise 11.2. Prove Proposition (c) A finite product of covering maps is a covering map.

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Q: Let A B and g: B-C be maps. Prove that if g of is one-to-one, then / is one-to-one.

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Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: Answer :

Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

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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: (a) Let X be a topological space. Let q: X→ A be a quotient map and let p: A → B be a surjection…

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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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