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)).

Q: Consider the relation R on the set of functions defined by (f, g) = R if and only if f(x) is…

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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…

A: It is given that X is the collection of all functions f: ℝ→ℝ. Show that there exists one-to-one…

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: 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: Provide two disjoint nonempty open sets C, D. Is it possible for a function g : R → R to be…

A: A set is a collection of elements like, numbers, objects. The numbers can be called as set of real…

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

A: See the attachment.

Q: continuous map со

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Q: • Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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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: Give an example (with justification) of a continuous function f : R→R such that for every non-empty…

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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: 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…

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: 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 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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