Exercise 7.14. Let X be a path-connected topological space. (a) Let f,g: I→ X be two paths from p to q. Show that f~g if and only if f.g~ Cp. (b) Show that X is simply connected if and only if any two paths in X with the same initial and terminal points are path-homotopic.

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A: Union of two topologies is not necessary a topology.

Q: (3) Consider the topological space (R. 7), where T= {UCR: U = 0 or R\U is countable}. (a) Show that…

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Q: Prove that 10. Let f(0) = 20 be the map of S periodic points of f %3D are dense in S.

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Q: 4. (a) Write down four distinct elements of the set {2 €C:2- E = 2i}.

A: Note:  We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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

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Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

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Q: 3. Show that is a closed subset of R². {(x, y) = R²: y = x²}

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Q: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

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Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…

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