Open Set — Definition, Formula & Examples

An open set is a set where every point inside it has some breathing room — you can move a small distance in any direction from any point and still remain within the set. The classic example is an open interval

(a, b)

, which includes all numbers between

a

and

b

but not the endpoints themselves.

A subset $U$ of a metric space $(X, d)$ is open if for every point $x \in U$ , there exists some $\varepsilon > 0$ such that the open ball $B(x, \varepsilon) = \{y \in X : d(x, y) < \varepsilon\}$ is entirely contained in $U$ . In general topology, open sets are the elements of a topology $\tau$ on a set $X$ , defined axiomatically: both $\emptyset$ and $X$ are in $\tau$ , arbitrary unions of sets in $\tau$ are in $\tau$ , and finite intersections of sets in $\tau$ are in $\tau$ .

Key Formula

\forall\, x \in U,\; \exists\, \varepsilon > 0 \text{ such that } B(x, \varepsilon) \subseteq U

Where:

$U$ = The set being tested for openness
$x$ = An arbitrary point in U
$\varepsilon$ = A positive real number (the radius of the open ball)
$B(x, \varepsilon)$ = The open ball of radius ε centered at x

How It Works

To check whether a set is open in

\mathbb{R}

, pick any point in the set and ask: can I find a small interval around this point that stays entirely inside the set? If the answer is yes for every point, the set is open. Endpoints of intervals fail this test — any interval centered at an endpoint will spill outside the set. The empty set

\emptyset

is open vacuously (there are no points to check), and

\mathbb{R}

itself is open because every real number has room around it within

\mathbb{R}

. Any union of open sets is open, and any finite intersection of open sets is open.

Worked Example

Problem: Show that the open interval (1, 5) is an open set in ℝ.

Pick an arbitrary point: Let x be any point in (1, 5). For example, take x = 3.

x = 3,\quad 1 < 3 < 5

Find a suitable ε: Choose ε to be the smaller of the two distances from x to the endpoints: the distance to 1 is 2, and the distance to 5 is 2. So set ε = 2 (or any smaller positive value).

\varepsilon = \min(x - 1,\; 5 - x) = \min(2, 2) = 2

Verify the open ball is contained in (1, 5): The open ball around x = 3 with radius 2 is the interval (1, 5), which is contained in (1, 5).

B(3, 2) = (1, 5) \subseteq (1, 5) \; \checkmark

Answer: For any x in (1, 5), setting ε = min(x − 1, 5 − x) guarantees B(x, ε) ⊆ (1, 5). Since this works for every point, (1, 5) is open.

Why It Matters

Open sets form the foundation of topology and real analysis. Continuity, convergence, and differentiability are all defined using open sets or their equivalent formulations. In optimization and economics, whether a feasible region is open or closed determines whether maximum and minimum values are actually attained.

Common Mistakes

Mistake: Assuming a set that is not open must be closed.

Correction: Sets can be neither open nor closed, such as [0, 1) in ℝ. A set can also be both open and closed (called clopen), like ∅ and ℝ itself.