Question 1. Let AcX. Show that t= {Uc X :ACU}U{ø} is a topology on X.

Q: Let X={a,b,c} and Y={1,2,3} and t={Y,¤,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

Q: Let E be a normed space and A CE be Ā = f" c0) for giver. some continuaus Show that function f: E -…

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Q: 2.2.15. Let X be a metric space, and suppose that xn → I and yn → K. Prove that d(xn, Yn) – + y in…

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Q: b) Which of the following maps are open? Give reasons for your answer. i) T:P' → P' given by…

A: By definition, a map f: X→Y is said to be open if, for any x∈X and every small neighborhood N of x,…

Q: Find the of im < Re z< £ un der w-

A: Substitute w=u+iv and z=x+iy in the transformation. w=1zu+iv=1x+iyu+iv=x-iyx+iy.1x-iyu+iv=x-iyx2+y2…

Q: Find the average rate of change of g(x) = – lr – 1 between the points (-1,0) and (1,-2) %3D

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

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Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: Q4) Let X={a, b, c, d} with the topology T={Ø, X, {a}, {b}, {a,b}, {b,c,d}}. Let f: (X,T)→(X,T) such…

A: Definition: Let f be a function defined from topological space X to topological space Y, then f is…