Question: Let f: X Y be function and g: X fcXxY, g(x) = (x, f(x)). Prove that if g is a homeomorphism then f is

Let f: X Y be function and g: X fcXxY, g(x) =

Let f: X Y be function and g: X fcXxY, g(x) = (x, f(x)). Prove that if g is a homeomorphism then f is continuous? Let g: Y Z be continuous and injective. Let f: X Y be a function. If gof is closed show that f is closed.

Step by Step Solution

★★★★★

3.49 Rating (156 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

be solution 1 let f xy function and g x f cx x ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Prove that if A is a diagonalizable matrix with only one eigenvalue , then a is of the form A = l. (Such a matrix is called a scalar matrix).

Prove that if S is a subspace of R1 either S = {0} or S = R1.

Prove that if A is a regular 2 Ã 2 matrix, then its L U factorization is unique. In other words, if where L, are special lower triangular and are upper triangular, then and A=LU = LU where L,L

4. Suppose that 'Ij;(B) and (E) are CP surfaces, and 'Ij; = 0 T, where T is a Cl function from B onto E. (a) If ('Ij;, B) and (, E) are smooth, and T is 1-1 with AT > 0 on B, prove for all...

Suppose that Ï{B) and Ï(E) are Cp surfaces and that Ï = Ï o Ï, where Ï is a C1 function from B onto E. a) If {Ï, B) and (Ï, E) are smooth, and Ï is 1-1...

Consider two queueing systems. The first has two servers; the service time for each is exponential with parameter . The second system has a single server whose service time is exponential with...

Let T = [0, 2). a) Prove that the function (x) := (cos x, sin x) is 1-1 from T onto B1(0, 0) R2. b) Prove that p(x, y) := || (x) - (y)|| is a metric on T. c) Prove that a function f is continuous on...

Let B denote the closed unit ball in a finite-dimensional normed linear space B = {x X: ||x|| 1} and let S denote its boundary, that is, S = {x X: ||x|| 1} There is no continuous function r: B S...

Let f be a convex function on an open set S that is bounded above by M in a neighborhood of x0; that is, there exists an open set f containing x0 such that f(x) ¥ M for every x U 1. Show that...

Let F be any function whose domain contains - x when-ever it contains x. Prove each of the following. (a) F(x) - F (- x) is an odd function. (b) F(x) + F(- x) is an even function. (c) F can always be...

Julie paid a day care center to watch her two-year-old son whileshe worked as a computer programmer for a local start-upcompany.What amount of child and dependent care credit can Julie claimin 202 2...

Conrad sends his quality control manager to Mexico to ensure the quality of a new manufacturer is meeting specifications. Which risk is Conrad addressing? Question 20 options: Process risk Control...

Exercise 5-4 Allocation of Cost and Workpaper Entries at Date of Acquisition On January 1, 2012, Porter Company purchased an 80% interest in Salem Company for $260,000. On this date, Salem Company...

One World Not Three (OWNT) Relief Fund is an international nonprofit relief agency based in the United States. OWNT recently celebrated its tenth anniversary and is in the middle of trying to expand...

explain why U.S. presidents often favor a "lower dollar" over the course of their presidency. Explain the relationship between the relative value of the dollar and its relationship to U.S. exports...

The real cost of palm oil case study discussed how the unsustainable farming of palm in Borneo, Sumatra and lower China is perhaps the biggest threat to _____________________. Group of answer choices...

On December 1 5 , Year 1 , the Binghamton Corporation established a retained earnings appropriation of $ 3 2 , 0 0 0 for future expansion. The balance of the retained earnings account prior to the...

From a medical tourist perspective, compare Shouldice with the traditional hospital in terms of the key factors of competition. Using Table 15-3, why would Shouldice attract patients from outside the...

Bert can swim at a rate of 2 miles per hour in still water. The current in a river is flowing at a rate of 1 mile per hour. If Bert wants to swim across the river to a point directly opposite, at...

Prove that T: Rn Rm is a linear transformation if and only if T (c1v1 + c2v2) = c1T(v1) + c2T (v2) For all v1, v2 in Rn and scalars c1, c2

Consider the subspace W of D, given by W = span (e2x, e-2x). (a) Show that the differential operator D maps W into itself. (b) Find the matrix of D with respect to B = {e2X, e-2x}. (c) Compute the...

Describe how an employee's benefit under a defined contribution plan is determined.

Why do employers use stock options in addition to salary to compensate their employees? For employers, are stock options treated more favorably than salary for tax purposes? Explain.

What are the differences between a traditional 401(k) and Roth 401(k) plan?