5. Consider the function f (X, J) → (Y, U), where (X, T) and (Y, U) are two topological spaces. Define f to be continuous. Then prove that the following statements are equivalent: (a) f is a continuous function; (b) Inverse image of every open set in Y is open in X; (c) Inverse image of every closed subset of Y is closed in X; (d) For any ACX, f(clA) C cl(f(A)).

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Q: R.2. Let X be the collection of all functions f :R→R. Show that there exists a one-to-one function G…

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Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

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Q: 3. Let f: X→ Y be a surjective, continuous map of topological spaces. Prove that if X is compact and…

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Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

Q: Describe the following sets in terms of graphs, supergraphs, and subgraphs of functions appropriate:…

A: let's break down each set step by step.Set A: {(x, y, z) | y^2 + x < yz ≤ x + y^2 + z^2}1. Graph…

Q: f: X → Y be a surjective function. Define a relation on X by setting ay iff f(x) = f(y). Show that…

A: Please check the solution in next step

Q: Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) 3D х is…

A: Suppose U is an open set in X. Its inverse image is then i−1(U) = {x ∈ A : i(x) ∈ U}          = {x…

Q: ► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a…

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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…

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Q: Prove that if R: R -> R is continuous, theis b2n the image of a bounded set over f is bounded

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Q: Suppose f: IR→ R and g: R → R and h: R→ R are continuous functions such that h(x) = f(x)g(x) for…

A: Using given information we have to tell that whether the  given statement is true or false.

Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

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Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: Show, using the definition of continuity from the topological point of view, that the constant…

A: See the attachment.

Q: Question 4. Let f:X→Y be a mapping from a topological space (X,rx) a topological space (Y,ry). Prove…

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Q: continuous map со

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Q: Let f:X → Y and g:Y → Z are functions show that a) If g of is surjective then g is surjective b) If…

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Q: (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.

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Q: Question 4. Let f:X Y be a mapping from a topological space (X,7x) a topological space (Y,Ty). Prove…

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Q: Question 4. Let f:X→Y be a mapping from a topological space (X,7x) a topological space (Y,ry). Prove…

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Q: Let f : Z → Z be relation defined by f(x) = ||æ| – 5|+ x for all æ E Z. i) f is a function ii) f is…

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Q: Theorem 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

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Q: 2. Find functions f,g: (0, 1) -+R with f continuous and g uniformly continuous such that neither fog…

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Q: Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) =…

A: Given, X, Y be topological spaces and y0∈Y. The constant map f:X→Y defined by fx=y0 Shows in the…

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

Q: 5.1: The following stat- X is disconnected.

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Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution

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Q: For functions:f(x, y) = x − yg(x, y) = x2 + x − yDetermine whether the following sets are closed and…

A: Final answer:.a) Not bounded.b) Not bounded.c) Not bounded.d) Not bounded.Explanation:Explanation:…

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

Q: Let f be a mapping of X into Y , A ⊆ X and B ⊆ Y . (a) Show that f(f−1(B)) ⊆ B and f−1(f(A)) ⊇ A.…

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Q: If f,g continuous on [a,b] and f(a) g(b). Show that there exists c ∈ (a,b) such that f(c) = g(c)

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Q: Prove that f is contraction map if and only if (X,τd)  to ( Y,τc)  is continuous

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