Please solve proposition 2.43 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. 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 E B, Vx E B1 N B2,3B3 E B, x E B3 C BỊN B2.%3DThe topology T on X generated by B is defined byT:= {UC X |3C C B,U = UC}.In other words, U eT 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 ET, for all x E U, there exists B E B such that x E B CU. Then B is a basisthat generates the topology T on X.

Name: Please solve proposition 2.43 in DETAIL. i posted this earlier and som
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Please solve proposition 2.43 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. Definition 2.41. A set B is a basis for a topology on a set X provided B satis