Union and Intersection of Open Sets¶

Proposition¶

For a metric space $(X,d)$ and a collection of open subsets $(E_i)_{i=1}^\infty$ of $X$ , the countable union $\displaystyle\bigcup_{i=1}^\infty E_i$ and the finite intersection $\displaystyle \bigcap_{i=1}^k E_i$ are also open.

Interpretation¶

Open sets are preserved under arbitrary unions and finite intersections. Infinite intersections can break openness, because the balls that fit inside the individual sets may shrink to nothing.

Open intervals from minus 1/n to 1/n for n = 1 to 4 stacked on number lines; their intersection over all n is the single point 0, which is not open.

The point $0$ is interior to every $G_n=(-\tfrac{1}{n},\tfrac{1}{n})$ $(- \frac{1}{n}, \frac{1}{n})$ , but the largest ball around $0$ inside $G_n$ has radius $\tfrac{1}{n}$ , and these radii tend to $0$ . No single ball around $0$ fits inside all $G_n$ , and indeed $\bigcap_{n\ge1}G_n=\{0\}$ is not open in $\mathbb{R}$ .

The proof is a statement about radii, using that a set is open if and only if all its points are interior points. A point of a union lies in some $E_j$ , and a ball around it inside $E_j$ also lies inside the union. A point of $E_1\cap\dots\cap E_k$ has balls of radius $\varepsilon_i$ inside each $E_i$ , and the smallest of finitely many positive radii, $\min_i\varepsilon_i>0$ $0$ , gives one ball inside all of them. With infinitely many sets the radii can have infimum $0$ , as in the figure.

Together with the openness of $\varnothing$ and $X$ , these two properties are the axioms of a topology. Taking complements gives the union and intersection of closed sets.

A common slip is to read the figure as "infinite intersections of open sets are never open". They may or may not be: $\bigcap_{n\ge1}(-n,n)=(-1,1)$ is open. The proposition only says that openness is not guaranteed.

Key Results¶

Both $X$ and $\varnothing$ are always open (and also closed).
In fact any union of open sets, not just countably many, is open. This holds for uncountable families as well.

Examples¶

In $\mathbb{R}$ , $\displaystyle \bigcup_{n=1}^\infty (0, 1-1/n) = (0,1)$ is open. Even though each interval misses the right endpoint, the union still does not include 1.
In $\mathbb{R}$ , $\displaystyle \bigcap_{n=1}^\infty (-1/n, 1+1/n) = [0,1]$ is closed. The intersection includes the endpoints 0 and 1, which are not interior points.

Prerequisites¶

Union and Intersection of Closed Sets

Sources¶

Math III: Analysis · Lecture 6

Symbols: epsilon, infinity, intersection, real numbers

Last updated: 20 October 2026 03:02