f is said to be an open mapping iff the image by f of any open set is open. Let TI be the lower limit topology generated by all unions of intervals of the form {[a,b):a,beR,as

Q: Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3),…

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

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Q: Let X:= (a, b.c), n=(1,2,3.4), Ty:= (0,Y, {1,3), [3,4}, {3), [1,3,4}}. Let fx-Y. f:= {(a,3). (b, 3).…

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Q: (a) f is open but neither closed nor continuous. (b) f is closed but neither open nor continuous.…

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Q: Let B {(x,y, z) E R: -1< y +x² < 1, 0 < z – r' < 2} CR. %3D (a) Prove that B is a closed subset of…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: A topological property is that which is preserved by homeomorphism.   ℝ is homeomorphic to (a, b).…

Q: Given: f = {(a,b),(b,a),(c,b)} a function from X = (a,b,c) to X a). Write f ∘ f as a set of…

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Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

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Q: Let f be a mapping from [1,+00[ to [1,+00[, defined by f(x)=x+1/x. Then * None of the choices f is a…

A: Given f is not defined at 0 but 0 is not in domain, so this does not create a problem.

Q: 2. Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?…

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Q: Let T = {0, 1, 2, 3} and the relation R is defined on T as follows: R = {(0, 1), (1, 2), (2, 2)}.…

A: The symmetric closure, R(s) is calculated using the formula, R(s)=R∪R-1 , where, R-1=b,a: a,b∈R .…

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 p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: Exercise 11.2. Prove Proposition (b) An injective covering map is a homeomorphism.

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Q: Which one of the following statements is true? * O Rwith the Euclidean topology and R with the…

A: Which of the following statement is true?

Q: Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (IR, 71) → (R,…

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Q: (e) On the other hand, for quotient maps : X→ X/T corresponding to actions of groups I' on a space…

A: e)Let's consider a quotient map ππ:X→X/T corresponding to the action of a group Γ on a space X. We…

Q: U xV, where U is an

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Q: The quotient topology actually defines a topology.

A: The quotient topology  actually defines a topology

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that…

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Q: Let A = {a, b, c, d}, B= {1, 2, 3}, and S= {(a, 1), (b, 1), (c, 2), (d, 3), (b, 3)}. a) Construct a…

A: We have to find the solution

Q: (e) A = (-1,1)U{2} in the standard topology on R %3D (f) A = (-1,1) U {2} in the lower limit…

A: Definition:(1) Limit points: Let (X, τ) be a topological space, a∈X is the limit pointof the set…

Q: (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.

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Q: Consider the space X {1, 2, 3} with the topolog Ty = {0, Y, {c}, the set Y {a, b, c} and f:X - Y is…

A: Image is attached with detailed solution.

Q: Let f be a mapping from [1,+[ to [1,+[, defined by f(x)=x+1/x. Then * O None of the choices O fis…

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Q: Construct a one-to-on continuous mapping of the set {(#1, 12, - ..., In); 0 < T; < 1. j= 1, 2,…

A: Given that : x1,x2,x3,.........,xn, 0&lt;xj&lt;1, j=1,2,3,.....,n So we have to take ,…

Q: f is said to be an open mapping iff the image by f of any open set is open. Let TI be the lower…

A:

Q: Let p:Y ->X be a covering map. Prove that any fiber is homeomorphic to a set with a discrete…

A: Let f:X--&gt;Y be a local homeomorphism. Suppose y∈Y and let x∈Fy: = f-1[{y}], the fibre of y. Then…

Q: Let f be a mapping from [1,+00[ to [1,+00[, defined by f(x)=x+1/x. Then * O None of the choices O…

A: The given data are: Let f be a mapping from ]0,+∞[ to [1,+∞[ by using the given function fx=1x.

Q: Let f be a mapping from [1,+ 0[ to [1,+°[, defined by f(x)=x+1/x. Then O None of the choices Ofis…

A: f(x)=x+1x Mapping [1,+∞) to [1,+∞) there is not a single point in the given domain [1,+∞) where…

Q: Prove that if f is a continuous mapping of a set S (in E) into F and g is a continuous mapping of…

A: Given