Suppose that {Va}aA is a collection of nonempty open sets in X which satisfies Va V

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Suppose that {Va}a∈A is a collection of nonempty open sets in X which satisfies Va ∩ Vβ = θ for all a ≠ β in A. Prove that if X is separable, then A is countable. What happens to this result when open is omitted?
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