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Boundary Point — Definition, Formula & Examples

A boundary point of a set SS is a point where every open neighborhood around it contains at least one point in SS and at least one point not in SS. The boundary point itself may or may not belong to SS.

A point xx is a boundary point of a set S⊆RnS \subseteq \mathbb{R}^n if for every ϵ>0\epsilon > 0, the open ball B(x,ϵ)B(x, \epsilon) satisfies B(x,ϵ)∩S≠∅B(x, \epsilon) \cap S \neq \emptyset and B(x,ϵ)∩Sc≠∅B(x, \epsilon) \cap S^c \neq \emptyset, where ScS^c denotes the complement of SS. The set of all boundary points of SS is denoted ∂S\partial S.

How It Works

To determine whether a point xx is a boundary point of SS, check whether every open interval (or open ball, in higher dimensions) centered at xx contains points both inside and outside SS. If you can find even one neighborhood that lies entirely within SS or entirely outside SS, then xx is not a boundary point. A set is closed if and only if it contains all of its boundary points, and open if and only if it contains none of them.

Worked Example

Problem: Determine the boundary points of the set S=(0,3]S = (0, 3] on the real number line.
Check x = 0: For any ϵ>0\epsilon > 0, the interval (−ϵ,ϵ)(-\epsilon, \epsilon) contains points less than 0 (not in SS) and points greater than 0 (in SS). So x=0x = 0 is a boundary point.
(−ϵ,ϵ)∩S≠∅and(−ϵ,ϵ)∩Sc≠∅(-\epsilon, \epsilon) \cap S \neq \emptyset \quad \text{and} \quad (-\epsilon, \epsilon) \cap S^c \neq \emptyset
Check x = 3: For any ϵ>0\epsilon > 0, the interval (3−ϵ,3+ϵ)(3 - \epsilon, 3 + \epsilon) contains points in SS (values slightly less than 3) and points not in SS (values greater than 3). So x=3x = 3 is a boundary point.
(3−ϵ,3+ϵ)∩S≠∅and(3−ϵ,3+ϵ)∩Sc≠∅( 3 - \epsilon, 3 + \epsilon) \cap S \neq \emptyset \quad \text{and} \quad (3 - \epsilon, 3 + \epsilon) \cap S^c \neq \emptyset
Check an interior point, say x = 1: Choose ϵ=0.5\epsilon = 0.5. The interval (0.5,1.5)(0.5, 1.5) lies entirely within SS, so x=1x = 1 is not a boundary point — it is an interior point.
(0.5,1.5)⊂S(0.5, 1.5) \subset S
Answer: The boundary of S=(0,3]S = (0, 3] is ∂S={0,3}\partial S = \{0, 3\}. Notice that 0∉S0 \notin S while 3∈S3 \in S, illustrating that boundary points may or may not belong to the set.

Why It Matters

Boundary points are central to understanding closed and open sets, which underpin rigorous definitions of limits, continuity, and compactness. In optimization, the Extreme Value Theorem guarantees a continuous function attains its maximum and minimum on a closed, bounded set — and those extrema often occur at boundary points.

Common Mistakes

Mistake: Assuming a boundary point must belong to the set.
Correction: A boundary point can lie inside or outside the set. For the open interval (0,1)(0,1), both 0 and 1 are boundary points even though neither belongs to the set.