Q. Let X,Y,Z,w be a topological spaces and f :X→Y, g:Y →Z , h:Z→ware continuous functions. Show that the composition function h•(gof):X-W is continuous.

Q: 6. (i) Let f: (X, J) → (Y, U), where (X, T) and (Y, U) are two topolog- ical spaces. Show that if…

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Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

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

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Q: Show that X'= B(X, K) is a Banach space.

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Q: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 12. Show that the set of all continuous function on the interval [-1,1] is a vector space under…

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Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Describe all linear maps f : R → R when R is regarded as an R-vector space

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Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: Let V = span{e2", xe2", x²e?=}. (a) Show that (a1e2a + a2xe2" + a3x²e2") e V for any a1, a2, az E R,…

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Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * O (X,T) is not connected but it…

A: No information is given for topology on Y So i have solved only the middle question

Q: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: ) Verify the continuity of the function f(x, y) 26– 2x -- y at the point (2,-3).

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Q: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: Find the value of b so that the point x = (13, b) belongs to the boundary of the set U = {x=R² : x₁…

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Q: Let (x, e) and (y, d) be metric space with e(x, y) = { 1 and let fr→y be a function. Is f…

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Q: Let f(x, y) = domain of f is the set D = {(x, y)|x² + y² > 1}. Select one: 1 1 - x² - y² True O…

A: Given.

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Q: Show that H = y:x+2y-z + 1 = 0 is not a sub-space of R³. -1=0} is

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Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: 1. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X,r) (Y,A) is continuous…

A: ANSWER :

Q: Let X and Y be two topological sPaces. Jet A and B CX be such that Ā =B. et f:X y be a continuous…

A: Given: X and Y be two topological spaces Let A and B ⊂X be such that A=B

Q: Let f(x,y) and let g(x, y) O rºy 26+y6 O 6x³y³ 6x6 + 6y g only Which of the above functions are…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: Let fi and f2 be concave functions from R" to R. Let f : R" +R be defined as the pointwise minimum…

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Q: X, Y be topological spaces and f : X Y be a continuous map. If X is completely normal then Y is also…

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