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The problem in the image asks you to show that the intersection of an arbitrary family of closed subsets of a topological space \( X \) is closed.

### Solution:
Let \( \{ A_\alpha \}_{\alpha \in I} \) be an arbitrary family of closed subsets of a topological space \( X \). We need to show that the intersection 
\[
\bigcap_{\alpha \in I} A_\alpha
\]
is closed in \( X \).

#### Definitions and Key Concepts:
1. **Closed Set**: A subset \( A \subseteq X \) is closed if its complement \( X \setminus A \) is open.
2. **Intersection of Sets**: The intersection of a family of sets \( \{ A_\alpha \}_{\alpha \in I} \) is the set of all elements that are in every \( A_\alpha \).

#### Proof:
To show that the intersection of closed sets is closed, we can use the fact that the complement of the intersection is the union of the complements.

Consider the complement of the intersection:
\[
X \setminus \left( \bigcap_{\alpha \in I} A_\alpha \right)
\]
Using De Morgan's laws, we have:
\[
X \setminus \left( \bigcap_{\alpha \in I} A_\alpha \right) = \bigcup_{\alpha \in I} \left( X \setminus A_\alpha \right)
\]
Since each \( A_\alpha \) is closed, \( X \setminus A_\alpha \) is open for each \( \alpha \in I \). The union of an arbitrary collection of open sets is open, so:
\[
\bigcup_{\alpha \in I} \left( X \setminus A_\alpha \right)
\]
is open in \( X \). Hence, the complement of the intersection is open, which means that the intersection itself:
\[
\bigcap_{\alpha \in I} A_\alpha
\]
is closed.

Thus, the intersection of any arbitrary family of closed subsets of a topological space \( X \) is closed.

### Do you have any questions or need further details?

### Related Questions:
1. What is the significance of De Morgan's laws in topology?
2. Can you provide an example where the intersection of an infinite number of closed sets is non-empty?
3. What happens if the family of sets is not closed? Can their intersection still be closed?
4. How do open sets relate to closed sets in a topological space?
5. What are the properties of arbitrary unions of open sets?

**Tip:** Always remember that in topology, understanding the relationship between open and closed sets is crucial for solving many problems.

Show that, if {A_α}_(α∈I) is an arbitrary family of closed subsets of X, then ⋂_(α∈I) A_α is closed.

This page provides a detailed proof showing that the intersection of an arbitrary family of closed subsets of a topological space X is closed. It explores key concepts in topology, including De Morgan's laws and properties of closed sets, suitable for undergraduate students.

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