Metric space. Unions and intersections of open sets

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 $\mathrm{\varnothing }$ and $X$ are open also.

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 $\mathrm{\varnothing }$ and $X$ are also closed.