Let f : (X,Tx)→ (Y,TY) be a function,where TX is a topology on X and TY is a topology on Y. a) Show that if the inverse image of every basic open set is open then the function f is continuous.

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: Let A and B be subsets of a set X. Prove that if A ⊆ Bc, then A and B are disjoint.

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A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

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A: This is a problem of topology.

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A: Please check the solution in next step

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Q: f(y). (a) Prove that is an (b) Find a bijection from A/~ to B. equivalence relation.

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Q: Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then…

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Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: Define a bijective map g : x2/3--->R.

A: Sol

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Q: On sets, the relation x R y given by there exists a bijective function from the set x to the set y'…

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Q: 3) Let R be the set of all real numbers with the euclidean topology and f: R → R be continuous…

A: Yes, the image of a continuous function from a path-connected space is always path-connected.…

Q: A: pp. A: px p. . A: pp(3)x3 + p(1)x.

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

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: Suppose that f : A → B and g : B → C are both injective functions, prove that the function go f : A…

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Q: Let the function f: R R be defined by f(r) = 3r -5. Prove that the function is bijective.

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Q: Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f: X → Y can be one-to-one or not…

A: By using definition of one-time and onto function we solve the given problem :

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: Use two of the following sets for each part below. Let X = {a,b,c}, Y = {1, 2, 3, 4} and Z = {s, t,…

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Q: Let A = {a, b, c, d, e}, B = {1, 2, 3, 4, 5} and let us consider the function f : A → B, f(a) = 3,…

A: Given that:  f : A → B, f(a) = 3, f(b) = 4, f(c) = 2, f(d) = 5, f(e) = 1.   Since, the images (i.e.…

Q: Let f:R→R with the usual topology that maps compact sets to compact sets in R, is the function…

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Q: Exercise 3 Prove that a function f: Rd - Rm is continuous if and only if f¹(U) C Rd is open for any…

A: Given:A function , where  represents the -dimensional Euclidean space and  represents the…

Q: A relation R on a set A is called irreflexive iff (Vx E A)((x, x) ¢ R). It is called asymmetric iff…

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Q: Suppose that R is a symmetric relation. Match the correct term with each blank space in the…

A: Suppose  that R is a symmetric relation than prove that R-1⊆R Proof: Suppose that <x,y>∈R-1.…

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Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

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Q: Let f be the subset of Z × Z defined by  f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z?…

A: Let f be the subset of Z × Z defined by  f = {(ab, a + b) : a, b ∈ Z}. We observe that  That is the…

Q: Let f : R → R and ECR. Prove f is continuous iff f-¹(E) is closed whenever ECR is closed.

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Q: Let R denote the set of real numbers. Let f: Rx R-> Rx R be a bijective function defined by f(x, y)…

A: Introduction :

Q: Let f : A → B be a function. Prove that f is surjective if and only if, for every pair of functions…

A: Please comment if you need any clarification. IF you find my answer useful put thumbs up. Thank you.

Q: 2. There is a function f : X →Y . Let's define a relation R in a set X, by a formula: (xRy) → (f(x)…

A: Given f: X → Y and R is the relation in the set X such that x R y ⇔ f(x) = f(y)

Q: Let A be a nonempty set and let f: A -> A be a function. Prove that if fof = iA, then f is…

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