How to prove that something forms a base topologically speaking

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To prove that the set of all open balls and the empty set form a base for a metric space (X,d), it is essential to demonstrate that their union covers the entire space X. The definition of a base requires that for every point in X and every neighborhood of that point, there exists an open ball contained within that neighborhood. Key properties to show include that the union of the open balls must equal X and that any open set can be expressed as a union of these balls. Clarifying these definitions and properties will help ensure that no critical aspects are overlooked in the proof. A thorough understanding of these concepts is crucial for successfully establishing the desired result.
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Homework Statement


If (X,d) is a metric space. I want to show that the set of all open balls and \emptyset form a base.

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The Attempt at a Solution


I know that we need to show that the union of all these sets (or balls) is the whole set. I feel like this is simple yet, I am unsure what to write so that I do not miss any important facts.
 
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Can you start by giving the definition of a base and the properties you need to show?
 
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