1. Let X and Y be topological spaces. Prove that a function f: X → Y is continuous if and only if f¹(C) is closed in X for every closed set CCY in Y.

Q: Theorem 8.10 (Intermediate Value Theorem). Let ƒ : Rstd map. If a, b e R and r is a point of R such…

A:

Q: 5. Consider the function f (X, J) → (Y, U), where (X, T) and (Y, U) are two topological spaces.…

A:

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 a normed space and let 0x0 EX. Show that there is f = X* such that f(x) = 1 and ||f|| =…

A: Given, a normed space  and .It is required to show that, there exists  (the dual space of ) such…

Q: Suppose that X and Y are topological spaces, and Y is a Hausdorff space. Suppose that A is a subset…

A: Given, X and Y are topological spaces, and Y is a Hausdorff space. Let A⊂X and A=X. Suppose that…

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: Let F(R, R) be the set of all functions from R to R. For any f, g € F(R, R), we define fog € F(R, R)…

A: Given is set of all functions from set of reals to set of reals

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…

A:

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: 3. Let X and Y be metric spaces. Let f :X → Y be a function. Definition: If f is discontinuous at a,…

A:

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 = {o, p, s, w}, Y = {a,b, e, g, k}, and Z = {1, 2, 3, 5, 7, 9}. Let ƒ : X → Y where f = {(o,…

A: X=o,p,s,wY=a,b,e,g,kZ=1,2,3,5,7,9f:X→Yf=o,a,p,g,s,e,w,bg:Y→Zg=g,3,e,2,b,9,k,7,a,1

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: continuous map со

A:

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

A:

Q: How do i prove 8.10? Could you include step by step explanation?

A: Consider the equation

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: 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.…

A:

Q: Prove that if f: R → R is continuous, then the image of a bounded set over f is bounded.

A: We need to prove that, if  is continuous, then the image of a bounded set over  is bounded.We know…