Math and science::Topology

# Metric space. Unions and intersections of open sets

- An arbitrary union of open subsets is open.
- A finite intersection of open subsets is open.
- Special cases: \( \emptyset \) and \( X \) are both open and closed.

### Lemma. Union and intersection of open sets

Let \( X \) be a metric space.

- Let \( (U_i)_{i \in I} \) be a family (finite or not) of open subsets of \( X \). Then \( \bigcup_{i\in I} U_i \) is also open in X.
- Let \( U_1 \) and \( U_2 \) be open subsets of \( X \). Then \( U_1 \cap U_2 \) is also open in X.
- Special cases \( \emptyset \) and \( X \) are open also.

The mirror statements for closed sets are as follows.

### Lemma. Union and intersection of closed subsets

Let \( X \) be a metric space.

- Let \( V_1 \) and \( V_2 \) be closed subsets of \( X \). Then \( V_1 \cup V_2 \) is also closed in X.
- Let \( (V_i)_{i \in I} \) be a family (finite or not) of closed subsets of \( X \). Then \( \bigcap_{i\in I} V_i \) is also open in X.

Special cases \( \emptyset \) and \( X \) are also closed.