Boundary Points Properties Quiz

1) Boundary points can be endpoints of the interval.

True

False

Endpoints like 0 and 1 in (0,1) or [0,1] meet the definition of boundary points.

Explanation

Endpoints like 0 and 1 in (0,1) or [0,1] meet the definition of boundary points.

2) The boundary of the interval [0,∞) is {0}.

True

False

Only 0 touches both the interval and the outside (negative numbers). Points >0 do not touch anything outside.

Explanation

Only 0 touches both the interval and the outside (negative numbers). Points >0 do not touch anything outside.

3) If A has no limit points, then the boundary of A is empty.

True

False

Boundary points are always limit points or isolated points that still touch the complement; if no limit points exist, there is no boundary.

Explanation

Boundary points are always limit points or isolated points that still touch the complement; if no limit points exist, there is no boundary.

4) If a point is in the closure of Aᶜ, it must be a boundary point of A.

True

False

A boundary point must be in both closure(A) and closure(Aᶜ).

Being only in the closure of Aᶜ is not enough.

Explanation

A boundary point must be in both closure(A) and closure(Aᶜ).

Being only in the closure of Aᶜ is not enough.

5) The boundary of an open interval (0,1):

(0,1)

{0,1}

∅

ℝ

The points 0 and 1 are not in the interval, but every neighborhood of 0 or 1 touches the interval and the outside, so the boundary is {0,1}.

Explanation

The points 0 and 1 are not in the interval, but every neighborhood of 0 or 1 touches the interval and the outside, so the boundary is {0,1}.

6) The boundary of a closed interval [0,1]:

(0,1)

{0,1}

∅

[0,1]

The endpoints 0 and 1 are where neighborhoods meet both the inside and outside of the interval, so the boundary is {0,1}.

Explanation

The endpoints 0 and 1 are where neighborhoods meet both the inside and outside of the interval, so the boundary is {0,1}.

7) The boundary of (1,∞):

ℝ

{1}

∅

(1,∞)

Only the point 1 is a boundary point because intervals around 1 touch both (1,∞) and numbers less than 1.

Explanation

Only the point 1 is a boundary point because intervals around 1 touch both (1,∞) and numbers less than 1.

8) Which statement best describes a boundary point of a set A?

A point only inside A

A point only outside A

A point where every neighborhood intersects both A and Aᶜ

A point with no neighborhood

A boundary point is one where every neighborhood touches both the set and its complement, meaning no matter how small the interval, it contains points inside A and outside A.

Explanation

A boundary point is one where every neighborhood touches both the set and its complement, meaning no matter how small the interval, it contains points inside A and outside A.

9) Boundary points of a set can be:

Only inside the set

Only outside the set

Inside or outside the set

Only limit points

A boundary point may lie inside the set or outside the set, as long as every neighborhood intersects both A and its complement.

Explanation

A boundary point may lie inside the set or outside the set, as long as every neighborhood intersects both A and its complement.

10) The boundary of ℝ:

∅

{0}

ℝ

Undefined

ℝ has no “outside,” so no point can touch both ℝ and its complement. The boundary is ∅.

Explanation

ℝ has no “outside,” so no point can touch both ℝ and its complement. The boundary is ∅.

11) A boundary point is always:

A limit point

An interior point

An isolated point

A maximum point

Boundary points are always limit points, since they require neighborhoods that always meet the set.

Explanation

Boundary points are always limit points, since they require neighborhoods that always meet the set.

12) A set with no interior points must have an empty boundary.

True

False

Example: ℚ has empty interior but boundary ℝ.

Explanation

Example: ℚ has empty interior but boundary ℝ.

13) A point can be both an interior point and a boundary point simultaneously.

True

False

Interior points require neighborhoods fully inside the set; boundary points require neighborhoods touching outside, so both cannot occur together.

Explanation

Interior points require neighborhoods fully inside the set; boundary points require neighborhoods touching outside, so both cannot occur together.

14) The boundary of the union of two sets is always the union of their boundaries.

True

False

This is not always true—taking the union can create new boundary points or remove some.

Explanation

This is not always true—taking the union can create new boundary points or remove some.

15) The boundary of the intersection of two sets is always the intersection of their boundaries.

True

False

This fails in general because intersections may change which points lie on the edge.

Explanation

This fails in general because intersections may change which points lie on the edge.