Question

Prove that A (AUB)' = 0.​

Answer 1

Answer:

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Answer 2

Answer:

To be more precise, let X be a topological space.

A and B are subsets of X. A′ and B′ are the derived sets of A and B, respectively. We have, (A∪B)′=A′∪B′.

(A∪B)′⊃A′∪B′ is trivial. To prove the converse, one has tried the following:

Let x∈(A∪B)' and let Ux a neighborhood of x.

Ux∖{x}∩(A∪B)≠∅.

In particular,

(Ux∖{x}∩A)∪(Ux∖{x}∩B)≠∅

and thus

Ux∖{x}∩A≠∅orUx∖{x}∩B≠∅.

Therefore x∈A'∪B'.

Step-by-step explanation:

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