prove the neighbourhood intersection lemma: the intersection of two neighbourhoods of x = * is itself a neighbourhood of x = *

To prove the *Neighbourhood Intersection Lemma*, we need to show that the intersection of two neighbourhoods of a point \( x \) is itself a neighbourhood of \( x \).

### Definitions:

- A set \( N \) is a *neighbourhood* of a point \( x \) in a topological space if there exists an open set \( O \) such that \( x \in O \subseteq N \). In other words, a neighbourhood of \( x \) contains an open set around \( x \).

- Let \( N_1 \) and \( N_2 \) be neighbourhoods of \( x \), meaning there are open sets \( O_1 \) and \( O_2 \) such that:
  \[
  x \in O_1 \subseteq N_1 \quad \text{and} \quad x \in O_2 \subseteq N_2.
  \]

We want to prove that \( N_1 \cap N_2 \) is a neighbourhood of \( x \).

### Proof:

1. Since \( N_1 \) is a neighbourhood of \( x \), there exists an open set \( O_1 \) such that:
   \[
   x \in O_1 \subseteq N_1.
   \]
   
2. Similarly, since \( N_2 \) is a neighbourhood of \( x \), there exists an open set \( O_2 \) such that:
   \[
   x \in O_2 \subseteq N_2.
   \]

3. Consider the open set \( O = O_1 \cap O_2 \). Since both \( O_1 \) and \( O_2 \) are open and contain \( x \), their intersection \( O \) is also open (because the intersection of open sets is open in a topological space), and it contains \( x \) (because \( x \in O_1 \) and \( x \in O_2 \)).

4. Now, observe that:
   \[
   O \subseteq N_1 \cap N_2.
   \]
   This is true because \( O \subseteq O_1 \subseteq N_1 \) and \( O \subseteq O_2 \subseteq N_2 \), so \( O \subseteq N_1 \cap N_2 \).

5. Therefore, we have found an open set \( O \) such that \( x \in O \subseteq N_1 \cap N_2 \), proving that \( N_1 \cap N_2 \) is a neighbourhood of \( x \).

### Conclusion:

The intersection of two neighbourhoods of \( x \) is itself a neighbourhood of \( x \), completing the proof.

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Would you like more details on any part of the proof or further explanations?

Here are 5 questions to expand on this:

1. How do the properties of open sets in a topological space affect neighbourhoods?
2. Can the intersection lemma be extended to the intersection of more than two neighbourhoods?
3. What happens if one of the neighbourhoods is not open?
4. How does the lemma apply to metric spaces specifically?
5. What other topological properties are preserved under set intersection?

**Tip:** Always remember that open sets are fundamental in defining neighbourhoods in topology, so understanding their properties is crucial.

Prove the neighbourhood intersection lemma: the intersection of two neighbourhoods of x = * is itself a neighbourhood of x = *.

This page provides a detailed proof of the Neighbourhood Intersection Lemma in topology, showing how the intersection of two neighbourhoods of a point is itself a neighbourhood. Key concepts include open sets and neighbourhoods, applicable for students at the undergraduate level.

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