Proof;
To prove cl(A∪B) = cl(A) ∪ cl(B) where A and B are Subsets of topological (X, T), we have need to show that cl(A) ∪ cl(B) ⊆ cl(A ∪ B) and cl(A ∪ B) ⊆ cl(A) ∪ cl(B).
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| Prove that if A and B are subsets of a topological space (X, T) then cl(A∪B) = cl(A) ∪ cl(B). |
As we know that
A ⊆ (A ∪ B) and B ⊆ (A ∪ B)
⇒ cl(A) ⊆ cl(A ∪ B) And cl(B) ⊆ cl(A ∪ B)
∵ if A ⊆ B ⇒ cl(A) ⊆ cl(B)
⇒ cl(A) ∪ cl(B) ⊆ cl(A ∪ B) ----> (1)
∵ if A ⊆ C and B ⊆ C ⇒ (A ∪ B) ⊆ C
⇒ (A ∪ B) ⊆ cl(A) ∪ cl(B)
∵ A ⊆ cl(A) and B ⊆ cl(B)
⇒ cl(A ∪ B) ⊆ cl(A) ∪ cl(B) --- > (2)
∵ cl(A) ∪ cl(B) is the closed super set of (A ∪ B) and cl(A ∪ B) is the smallest closed super set of (A ∪ B)
From (1) and (2) we have
cl(A ∪ B) = cl(A) ∪ cl(B)