The Central Idea
Given a set or spatial object \(S\), points around it can often be classified by how their immediate surroundings relate to \(S\).
- An interior point lies fully inside \(S\).
- An exterior point lies fully outside \(S\).
- A boundary point lies at the transition between \(S\) and its complement.
- A neighborhood describes the points sufficiently close to a chosen point.
These ideas help us distinguish an object from its edge and understand what happens locally around individual points.
Why Local Spatial Language Matters
Two points can both belong to the same region but behave very differently.
One may lie deep inside the region, while another lies directly on its edge.
Likewise, two points outside the region may differ: one may be just beyond the boundary, while another may be very far away.
Neighborhood language lets us study what happens close to a point rather than only asking whether the point belongs to the set.
Interior
The interior of a set consists of the points that lie fully inside the set.
Informally, an interior point has some amount of room around it that remains entirely inside the set.
If \(p\) is an interior point of \(S\), then a sufficiently small neighborhood around \(p\) is contained in \(S\).
Interior of an Interval
Consider the closed interval
\[ [2,7]. \]The points strictly between \(2\) and \(7\) are interior points.
Therefore, its interior is
\[ \boxed{(2,7)}. \]The endpoints belong to the interval but are not interior points.
Interior of a Disk
Consider the closed disk
\[ D=\{(x,y):x^2+y^2\le 9\}. \]Its interior consists of the points satisfying
\[ \boxed{x^2+y^2<9}. \]Every such point has a sufficiently small disk around it that remains inside \(D\).
Interior Does Not Mean Center
An interior point does not have to be near the center of a figure.
A point very close to the boundary can still be interior as long as some sufficiently small neighborhood around it remains entirely inside the set.
Interior is a local property, not a measure of centrality.
Exterior
The exterior of a set consists of points that lie fully outside the set.
Informally, an exterior point has some small neighborhood containing no points of the set.
Thus, exterior points are not merely points that fail to belong to the set. Boundary points may also fail to belong to a set without being exterior.
Exterior of a Disk
For the closed disk
\[ D=\{(x,y):x^2+y^2\le9\}, \]the exterior is
\[ \boxed{x^2+y^2>9}. \]These points lie strictly beyond the circular boundary.
Exterior Depends on the Ambient Space
The meaning of exterior depends on the larger space in which the set is being considered.
A line segment viewed inside a line behaves differently from the same segment viewed inside the entire plane.
This broader surrounding space is sometimes called the ambient space.
Later mathematics makes this dependence explicit.
Boundary
The boundary of a set consists of points where the set and its outside meet.
A boundary point has the following intuitive property:
Every sufficiently small neighborhood around it contains points of the set and points outside the set.
Boundary of an Interval
For the interval
\[ [2,7], \]the boundary consists of the two endpoints:
\[ \boxed{\{2,7\}}. \]Near either endpoint, one can find points inside the interval and points outside it.
Boundary of a Disk
For
\[ D=\{(x,y):x^2+y^2\le9\}, \]the boundary is the circle
\[ \boxed{x^2+y^2=9}. \]Every small neighborhood around a point on this circle contains points inside the disk and points outside it.
A Boundary May Belong to the Set
Boundary points are not automatically outside a set.
In the closed disk
\[ x^2+y^2\le1, \]the boundary
\[ x^2+y^2=1 \]is included in the set.
A Boundary May Be Excluded from the Set
Consider the open disk
\[ x^2+y^2<1. \]Its boundary is still
\[ x^2+y^2=1, \]but none of those boundary points belong to the open disk.
Thus, whether a boundary belongs to a set and whether a point is a boundary point are separate questions.
The Same Boundary Can Belong to Different Sets
The open disk
\[ x^2+y^2<4 \]and the closed disk
\[ x^2+y^2\le4 \]have the same boundary:
\[ \boxed{x^2+y^2=4}. \]The difference is whether the boundary is included in the set.
Neighborhood
A neighborhood of a point is a collection of points near that point.
In Euclidean space, a standard neighborhood of point \(p\) with radius \(r>0\) is the open ball consisting of points whose distance from \(p\) is less than \(r\).
Symbolically,
\[ \boxed{ B_r(p)=\{x:d(x,p)Neighborhoods on the Number Line
On the real number line, an \(r\)-neighborhood of \(a\) is
\[ \boxed{(a-r,a+r)}. \]For example, a neighborhood of radius \(2\) around \(5\) is
\[ (3,7). \]Neighborhoods in the Plane
In the Euclidean plane, an \(r\)-neighborhood around
\[ P=(h,k) \]is the open disk
\[ \boxed{ (x-h)^2+(y-k)^2Neighborhoods in Three Dimensions
In three-dimensional Euclidean space, an \(r\)-neighborhood around
\[ P=(a,b,c) \]is the open ball
\[ \boxed{ (x-a)^2+(y-b)^2+(z-c)^2Interior Points and Neighborhoods
A point \(p\) is an interior point of \(S\) when there exists some radius \(r>0\) such that
\[ \boxed{B_r(p)\subseteq S}. \]The radius does not need to be large.
It only needs to be positive.
Exterior Points and Neighborhoods
A point \(p\) is exterior to \(S\) when some neighborhood around \(p\) contains no points of \(S\).
Equivalently, for some \(r>0\),
\[ \boxed{B_r(p)\cap S=\varnothing}. \]Boundary Points and Neighborhoods
A point \(p\) is a boundary point of \(S\) when every neighborhood around \(p\) meets both \(S\) and its complement.
Symbolically, every \(r>0\) satisfies
\[ B_r(p)\cap S\neq\varnothing \]and
\[ B_r(p)\cap S^c\neq\varnothing. \]This is the precise local meaning of being “on the edge.”
Interior, Boundary, and Exterior Partition Space
For many familiar subsets of Euclidean space, every point can be classified as interior, boundary, or exterior relative to the set.
These categories do not overlap.
Informally,
\[ \text{space} = \text{interior} \cup \text{boundary} \cup \text{exterior}. \]This gives a useful organizational picture for spatial classification.
Open Sets
A set is open when every point of the set is an interior point.
For example,
\[ (2,7) \]is open in the real number line.
Likewise,
\[ x^2+y^2<9 \]describes an open disk in the Euclidean plane.
Closed Sets
A set is closed when it contains all of its boundary points.
For example,
\[ [2,7] \]is closed in the real number line.
Similarly,
\[ x^2+y^2\le9 \]is a closed disk in the Euclidean plane.
Open and Closed Are Not Opposites in Ordinary Language
In topology, open and closed are technical properties.
A set can be both open and closed in a given space.
For example, the entire real number line
\[ \mathbb{R} \]is both open and closed as a subset of itself.
The empty set has the same property.
The Boundary of an Open Set Need Not Belong to It
The interval
\[ (0,1) \]is open.
Its boundary is
\[ \{0,1\}. \]Neither boundary point belongs to the interval.
The Boundary of a Closed Set Belongs to It
The interval
\[ [0,1] \]is closed.
Its boundary is again
\[ \{0,1\}, \]and both endpoints belong to the set.
A Set Can Have Empty Interior
A line in the plane has no two-dimensional interior.
No matter how small a disk is centered at a point on the line, that disk contains points not on the line.
Therefore, viewed as a subset of the plane, a line has empty interior.
Dimension Affects Interior
Whether an object has interior depends on the surrounding space.
A line segment has interior relative to the line containing it, but it has no interior when viewed as a subset of the entire plane.
This distinction is called relative interior in more advanced mathematics.
The Boundary Can Be the Entire Set
A lower-dimensional object embedded in a higher-dimensional space can consist entirely of boundary points.
For example, a line viewed as a subset of the plane has empty interior, and every point on the line is a boundary point.
Thus, its boundary in the plane is the entire line.
Interior and Exterior Are Open
In Euclidean space, the interior of a set is open.
Its exterior is also open.
This follows from the neighborhood definitions: interior and exterior points each have some small neighborhood that stays entirely within the same classification.
The Boundary Is Closed
The boundary of a set is closed.
Intuitively, taking limits of boundary points cannot suddenly produce a point that is safely inside or safely outside without encountering the boundary behavior nearby.
Formal proofs of this statement belong to topology, but the geometric idea is already useful here.
Closure
The closure of a set is obtained by including all of its boundary points.
It is commonly written
\[ \overline{S}. \]For example,
\[ \overline{(2,7)}=[2,7]. \]The closure combines the interior or existing set with its limiting edge points.
Closure of an Open Disk
If
\[ D=\{(x,y):x^2+y^2<4\}, \]then its closure is
\[ \boxed{ \overline{D} = \{(x,y):x^2+y^2\le4\} }. \]The circular boundary is added.
Boundary from Closure and Interior
A useful advanced identity is
\[ \boxed{ \partial S = \overline{S}\setminus\operatorname{int}(S) }. \]Here,
\[ \partial S \]denotes the boundary of \(S\).
The boundary is what remains after the interior is removed from the closure.
Neighborhoods Need Not Be Circular in Every Geometry
Circular disks and spherical balls arise from ordinary Euclidean distance.
If a different distance rule is used, neighborhoods can have different shapes.
For example, under the taxicab metric,
\[ d((x_1,y_1),(x_2,y_2)) = |x_2-x_1|+|y_2-y_1|, \]neighborhoods appear diamond-shaped in Cartesian coordinates.
The concept of neighborhood is more general than any particular drawing.
Neighborhoods Express Local Behavior
Many later mathematical ideas ask what happens in every sufficiently small neighborhood of a point.
This language supports:
- continuity;
- limits;
- local extrema;
- connectedness;
- topological equivalence;
- local geometry.
The neighborhood concept is therefore one of the bridges between geometry and analysis.
Interior and Constraints
In optimization, a feasible region may contain interior points and boundary points.
For example,
\[ x\ge0,\qquad y\ge0,\qquad x+y\le10 \]defines a closed triangular feasible region.
Points satisfying all three inequalities strictly,
\[ x>0,\qquad y>0,\qquad x+y<10, \]lie in its interior.
Boundary and Equality Constraints
Boundaries often arise when an inequality becomes an equality.
For
\[ x+y\le10, \]the line
\[ x+y=10 \]forms part of the boundary of the corresponding half-plane.
This algebraic pattern appears repeatedly in geometry and optimization.
Interior of a Polygonal Region
The interior of a polygonal region consists of points lying inside all of its boundary edges.
The edges and vertices themselves form the boundary.
If the region includes its edges, it is a closed polygonal region.
If the edges are excluded, the interior alone forms an open region relative to the plane.
Interior of a Solid
For a solid ball
\[ x^2+y^2+z^2\le r^2, \]the interior is
\[ \boxed{x^2+y^2+z^2is the boundary.
Exterior of a Solid
For the same solid ball, the exterior consists of points satisfying
\[ \boxed{x^2+y^2+z^2>r^2}. \]Interior, boundary, and exterior therefore extend naturally from the plane into three-dimensional space.
Common Error: Assuming Every Point in a Set Is Interior
Boundary points can belong to a set without being interior points.
In
\[ [0,1], \]the point \(0\) belongs to the interval but is not interior.
Every neighborhood of \(0\) contains real numbers less than \(0\), which lie outside the interval.
Common Error: Assuming Every Point Outside a Set Is Exterior
A boundary point may be excluded from a set without becoming an exterior point.
For the open interval
\[ (0,1), \]the point \(0\) does not belong to the set.
But every neighborhood of \(0\) contains points of \((0,1)\), so \(0\) is a boundary point, not an exterior point.
Common Error: Assuming Boundary Means Included
Boundary describes local position relative to a set.
Inclusion is a separate membership question.
Closed sets contain their boundaries. Open sets generally do not.
Common Error: Treating Open and Closed as Exact Opposites
A set can be neither open nor closed.
For example,
\[ [0,1) \]is neither open nor closed in \(\mathbb{R}\).
A set can also be both open and closed, as with \(\mathbb{R}\) considered inside itself.
Common Error: Forgetting the Ambient Space
The same object can have different interior and boundary behavior depending on the larger space in which it is considered.
A segment has nonempty relative interior along its supporting line but empty interior in the full plane.
Always identify the surrounding space when precision matters.
Common Error: Thinking Neighborhood Means Nearest Point
A neighborhood is not a single nearby point.
It is an entire collection of points surrounding a chosen point according to some distance or topological rule.
A Reliable Classification Procedure
To classify a point \(p\) relative to a set \(S\), ask:
- Does \(p\) belong to \(S\)?
- Can I place a sufficiently small neighborhood around \(p\) entirely inside \(S\)?
- If yes, \(p\) is an interior point.
- Can I place a sufficiently small neighborhood around \(p\) entirely outside \(S\)?
- If yes, \(p\) is an exterior point.
- If every neighborhood contains both points of \(S\) and points outside \(S\), then \(p\) is a boundary point.
A Reliable Procedure for Inequality-Defined Regions
- Identify the equality case; it often describes the boundary.
- Determine which side of the boundary satisfies the inequality.
- Use strict inequalities to identify interior portions when appropriate.
- Check whether equality is included.
- Classify points by substitution when possible.
A Compact Comparison
| Idea | Local description | Disk example |
|---|---|---|
| Interior | Some neighborhood lies entirely in the set | \(x^2+y^2<r^2\) |
| Boundary | Every neighborhood meets the set and its complement | \(x^2+y^2=r^2\) |
| Exterior | Some neighborhood lies entirely outside the set | \(x^2+y^2>r^2\) |
| Neighborhood | Points sufficiently close to a chosen point | \((x-h)^2+(y-k)^2<\rho^2\) |
Practice 1: Interval Interior
Find the interior of
\[ [-3,5]. \]Practice 2: Interval Boundary
Find the boundary of
\[ (-2,6). \]Practice 3: Disk Interior
Find the interior of
\[ x^2+y^2\le25. \]Practice 4: Disk Exterior
Find the exterior of
\[ x^2+y^2\le25. \]Practice 5: Classify a Point
Classify the point
\[ P=(3,4) \]relative to the disk
\[ x^2+y^2\le25. \]Practice 6: Neighborhood on the Number Line
Write the radius-\(3\) neighborhood of the point \(8\) on the real number line.
Practice 7: Neighborhood in the Plane
Write an inequality for the radius-\(2\) neighborhood centered at
\[ P=(1,-1). \]Practice 8: Open or Closed?
Classify
\[ x^2+y^2<16 \]as open or closed in the Euclidean plane.
Practice 9: Boundary Included?
Does the set
\[ x^2+y^2<9 \]contain its boundary?
Practice 10: Identify the Error
A student says, “If a point does not belong to a set, then it must be an exterior point.”
Explain why this is false.
Practice Answers
Practice 1
\[ \boxed{(-3,5)}. \]Practice 2
The boundary is
\[ \boxed{\{-2,6\}}. \]Practice 3
\[ \boxed{x^2+y^2<25}. \]Practice 4
\[ \boxed{x^2+y^2>25}. \]Practice 5
Compute
\[ 3^2+4^2=9+16=25. \]Therefore, \(P\) lies on the boundary of the disk.
Practice 6
\[ (8-3,8+3) = \boxed{(5,11)}. \]Practice 7
\[ \boxed{(x-1)^2+(y+1)^2<4}. \]Practice 8
The set is open because every point satisfying the strict inequality has a sufficiently small neighborhood remaining inside the set.
Practice 9
No. Its boundary is
\[ x^2+y^2=9, \]and those points do not satisfy the strict inequality.
Practice 10
A point can fail to belong to a set and still be a boundary point. For example, \(0\notin(0,1)\), but every neighborhood of \(0\) contains points from \((0,1)\). Therefore, \(0\) is a boundary point rather than an exterior point.
What to Remember
- Interior points have some neighborhood entirely inside the set.
- Exterior points have some neighborhood entirely outside the set.
- Boundary points have every neighborhood meeting both the set and its complement.
- A boundary point may belong to the set or may be excluded from it.
- Open sets contain only interior points.
- Closed sets contain all of their boundary points.
- Open and closed are technical properties, not simple opposites.
- Neighborhoods make the everyday idea of “nearby” mathematically precise.
- Interior and boundary depend on the ambient space.
- Equations often describe boundaries, while strict and non-strict inequalities distinguish interiors and closed regions.
Where We Go Next
Interior, exterior, boundary, and neighborhood describe how points relate to the local structure of spatial objects. The next step is to compare objects according to sameness of shape, scale, structure, and transformation.
The next lesson, Congruence, Similarity, Equivalence, and Symmetry, develops several different meanings of geometric sameness and the transformations that preserve them.