Let X = {a,b,c,d} and Tx = {X, Ø, {a}, {a, b}, {a,b,c}} be a topology on X. Let y = {1,2,3,4} and Ty = {Y, Ø, {1}, {1,3,4}} be a topology on Y. Let f be a mapping of X into Y defined by: f(a) = f(c) = 1, f(b) = 2 and f(d) = 3. Let g be a mapping of X into Y defined by: g(a) = 1, g(b) = g(c) = 4 and g(d) = 2. Then: Of is not continuous and g is continuous O f and g are continuous Of and g are not continuous. Of is continuous and g is not continuous

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: Let X = {a, b, c, d} with the following topology: ?= {X,0, {a},{b}, {a,b}} Let Y={x,y,z} Consider…

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Q: Let A and B be topological spaces and endow A x B and B x A with respective product topologies. Show…

A: Homeomorphism: Let X and Y be topological spaces; let f:X→Y be a bijection. If both f and f-1:Y→X…

Q: 1. Let A = {a, b, c}, B = {1,2,3}, C = {x, y, z, w}, D = {4,5, 6}. Let the functions f : A → B, g :…

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Q: be a topology on Y. Let f be a mapping from X into Y defined by f(a)=f(b) = 3, f(c = 2 and f(d) = 1.…

A: Continuity of functions

Q: Let C = {0, 1, 2, 3} and f: C → C, g: C → C with ƒ = {(0, 2), (1, 2), (2, 3), (3, 0)}, {(0, 3), (1,…

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Q: Let a relation S be defined on A = {0, 1, 2, 3} as follows. S= {(0, 0), (0, 3), (1, 0), (1, 2), (2,…

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Q: Let X and U be Banach spaces, let C: X→ U be a compact linear map, and let xnx in X. Show that Can →…

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Q: Let X:= {a, b, c}, Y:= {1,2,3,4}, Ty: = {0, Y,{1,3}, {3,4}, {3}, {1,3,4}}. Let f:X → Y, f:= {(a, 3),…

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Q: Let X: = {a, b, c}. Y: = {1,2,3,4}. Let f:X → Y, f:= {(a, 3), (b, 3), (c, 4)}. If Ty = {Ø, X, {a,…

A: Let x:=a,b,c   y:=1,2,3,4   Let f:x→yf:=a,3,b,3,c,4  if Tx=ϕ,x,a,b,aTy:=ϕ,y,1,33,431,3,4   then f…

Q: Let A={1,2,3,4} and let R, S, T, and U be the following relations: R={(1,1),(1,2),(1,3),(1,4)}…

A: Given :  A={1,2,3,4} and let R, S, T, and U be the following relations: R={(1,1),(1,2),(1,3),(1,4)}…

Q: Let X = {a,b,c, d},T= {0,X, {d}, (a,b}, {a,b, d}} and Y= {m,n,P. q},F = {¢,Y, {p}, {m,n}, {m,n,p}}…

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Q: Consider the relation R on the set {a,b,c,d,e,f} which is defined as follows: R={(a,a), (a,b),…

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Q: Let W = (0, 1, 3}, X = {1, 3, 5), Y = (3, 4} and Z = {(1, 2), (3, 4), (1, 3),(2, 4). Compute (X x Y)…

A: The given problem is to find the Cartesian product of given sets and verify the Cartesian product…

Q: 1. Let V = F∞ and consider the two linear maps T, S = L(F∞, F∞ defined by T(a₁, A2, A3, .) = (a2,…

A: Let  and consider  the  two linear  maps defined  by…

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 = {a, b,c, d},T= {o, X, {d}, {a,b}, {a, b, d}} and Y = (m,n, p,g},F = {@,Y, {p}, (m,n}, {m,n,…

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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: Let X = {1,2,3,4,5} and let T = = {x, Ø, {2,3,4,5}, {1,2,5}, {2,5}, {1}}. O (X,T) is Hausdorff but…

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Q: {X, Ø, {1}, {2},{1,2},{1,3}}, is (X, t) a connected a) Let X = {1,2,3} with the topology t=…

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Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

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Q: Let X = {0, r, t, w}, Y = {e,g, i, k, m}, and Z = {2,3,4,5, 7,9}. Let f : X → Y where f = {(0, g),…

A: Given X = {o, r, t, w} Y = {e, g, i, k, m} Z = {2, 3, 4, 5, 7, 9}

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: 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, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

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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: 2. Let f A B be a mapping. Prove that f is bijective if and only if f(A\C) = B\ƒ(C) for all CCA.

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Q: Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3,…

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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: Let π D₁ = {z € C\{0} : < Arg z < 4 D₂ = {z € C: 0 ≤ Re(2) ≤ ½ & 0 ≤ Im(2) ≤ 1}. a) Sketch D₁ and D₂…

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Q: (5.8) Symmetric Bilinear Maps: Let f:V x V F is bilinear map, then f is called symmetric if f(A, B)…

A: Example of such bilinear map is  the standard dot product is a symmetric bilinear form, B(x, y) = x…

Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

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Q: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

A: Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this…

Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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