Open sets form a topology

Open sets form a topology: Let $(X,d)$ be a metric space . Then:

• $\varnothing$ and $X$ are open ;
• arbitrary unions of open sets are open: if $\{U_\alpha\}_{\alpha\in A}$ are open, then $\bigcup_{\alpha\in A} U_\alpha$ is open;
• finite intersections of open sets are open: if $U_1,\dots,U_n$ are open, then $\bigcap_{j=1}^n U_j$ is open.

These closure properties justify treating “open sets” as the primitive objects defining the topological structure induced by a metric .

Proof sketch (optional): Use the $\varepsilon$-ball definition: for unions pick the ball guaranteed by the open set containing the point; for intersections use the minimum of the radii from each open set.