Q2: a) Let f be a map from a set X iato a space (Y,T) and T'- {Uc xaV eT, U=f"(vshow that T is smallest topology on X,makes f continuous. b) Let T be a co finite topology on a set of reai numbers R and f:R →R be a mnap Sueh that fx) =-k+ 2, show that whether f is T-T continuous or not.

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

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Q: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

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Q: (C0]) }

A: The statement is false. Consider f:[0,1]→ℝ defined as f(x)=tan-1πx2   x∈[0,1)0              x=1

Q: f) Every continuous map T² RP2 is homotopic to the constant map.

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A: Given that,f(z)=z+1zz=1

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A: Given: v) To prove the given statement as follows,

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Q: 1) Let S be the equivalence relation on R defined by xSy iff x-y Q. Then a) [√3]-(-√3+r,req) b)…

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Q: 5. Let X =[0,2). Define T = {[0,a):0<a<2}. %3D a) Show that T is a topology on X. b) What are the…

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Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: Let f be the map sending each Comp lex num bey z> Show f(2,Z2) = f (2,) f (2») %3D

A: The detailed solution is as follows below :

Q: #4. Set XyZ F(x4a)= | 4 3ゃex xy tyz+zX メ*ytを Show that F is 11 an open set ontaining (1,0,-1). in O…

A: Given - Let Fx, y, z = xyzxy + yz + zxx + y + z To Show -  Show that F is 1 : 1 in an open set…

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A: Please to the answer in the next step

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Q: Question 3. If X and Y are isometric and X is complete, show that Y is complete.

A: Given below the detailed solution

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Q: Consider the topology T on X= (a, b, c, d, e), (X, Ø, (a), (e,d}, {a, e, d), (b, c, d, e}} Show that…

A: It is given that,  X = {a, b, c, d, e} and  T = X, ∅, a, c, d, a, c, d, b, c, d, e be a topology on…

Q: Let f:X → Y is a function and let {Ba: a e A} be an indexed family of Subset of Y. Show that a)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2- i) Let B {(x,y):x,y eZ and 3(x-y)}. Show that B is an equivalence relation. Find equivalance…

A: A relation is said to be equivalence relation of it is reflexive, symmetric and transitive.…

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

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Q: Q2. Answer as true or false: (a) f-'((AUB)nC) = (ƒ-'(A)n f-1C)) U (f-1(B) n f-1(C)). (b) Given the…

A: (a) is True. (b) is False.

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Q: Consider the relation R = {(x,x): x € Z} on Z. Is R reflexive? Symmetric? Transi- tive? If a…

A: We have to check given relation R for reflexive,symmetric and transitive. Concept required: R:A→B is…

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Q: (c) let t be indiscrete topology on set of integers Z, show that whether (Z, t) is regular space or…

A: Given : ℤ , τ is indiscrete topological space. To show : To show whether ℤ , τ is regular , T0, T2,…

Q: 5) Let 7 be the euclidean topology on R and 7' be the finite-closed topology on R. Define f : (R,…

A: