Please solve THEOREM 2.47 in DETAIL. i posted this earlier and someone posted the wrong answer from CHEGG, so please dont do that. only attempt if you know the answer. 2.5. Basis for a Topology. Sometimes a topology on a set is too complicated to list allthe open sets. In that case, we use a smaller collection of open sets, called a basis to generatethe desired topology.Exercise 2.40. You have already seen this idea in a few previous exercises, examples, andproblems. Which ones?Definition 2.41. A set B is a basis for a topology on a set X provided B satisfies thefollowing axioms:(1) UB = X.(2) VB1, B2 € B,Vx € B1N B2,3B3 € B,x € B3 C BỊn B2.The topology T on X generated by B is defined byT:= {U c X | 3C C B,U = UC}.In other words, U €T iff U is a union of elements of B.Exercise 2.42. Show that: (1) The topology generated by a basis is a topology. (2) Thesets in the basis are open sets in that topology.Proposition 2.43. Let X be a space with topology T and B a subcollection of T such thatfor all U E T, for all x e U, there erists B e B such that x € B C U. Then B is a basisthat generates the topology T on X.We can do it even cheaper!Definition 2.44. A set S is a subbasis for a topology on a set X provided US = x.The topology I on X generated by S is defined to be the collection of all unions of finiteintersections of elements of S.Problem 2.45. Can you define the basis generated by a subbasis?Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) Thesets in the subbasis are open sets in that topology.Theorem 2.47. Suppose B is a basis for a topology on X and Y c X. Then By := {BnY ||BE B} is a basis for the subspace topology on Y.

Name: Please solve THEOREM 2.47 in DETAIL. i posted this earlier and someone
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Description: Please solve THEOREM 2.47 in DETAIL. i posted this earlier and someone posted the wrong answer from CHEGG, so please dont do that. only attempt if you know the answer. 2.5. Basis for a Topology. Sometimes a topology on a set is too complicated to lis