Proof;
To prove cl(A∩B) ⊆ cl(A)∩cl(B), where A and B are Subsets of X, we have need to show that (A∩B) ⊆ cl(A) ∩ cl(B) at first and then 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). |
We know that A ⊆ cl(A) and B ⊆ cl(B).
⇒ (A ∩ B) ⊆ cl(A) and (A∩B) ⊆ cl(B)
⇒ (A ∩ B) ⊆ cl(A) ∩ cl(B)
⇒ cl(A ∩ B) ⊆ cl(A) ∩ cl(B)
∵ cl(A) ∩ cl(B) is closed super set of (A ∩ B) and cl(A ∩ B) is the smallest closed super set of (A ∩ B)
Hence proved that cl(A ∩ B) ⊆ cl(A) ∩ cl(B), where A and B are Subsets of topological Space (X, T).