Open Balls: Concept Mastery Quiz

1) Every open ball B(x,r) in a metric space contains its center x.

True

False

True, because d(x,x)=0 0, so the center is always included in the open ball.

Explanation

True, because d(x,x)=0 0, so the center is always included in the open ball.

2) In a metric space (X,d), the open ball B(x,r) is defined as:

{y ∈ X : d(x,y) ≤ r}

{y ∈ X : d(x,y) < r}

{y ∈ X : d(x,y) = r}

{y ∈ X : d(x,y) > r}

By definition, an open ball uses strict inequality: all points whose distance from x is strictly less than r.

Explanation

By definition, an open ball uses strict inequality: all points whose distance from x is strictly less than r.

3) For any metric d, an open ball is always a nonempty set.

True

False

True, because every open ball contains at least its center, so it can never be empty.

Explanation

True, because every open ball contains at least its center, so it can never be empty.

4) In the usual metric on ℝ, the open ball B(3,4) is:

[3−4, 3+4]

(3−4, 3+4)

(3,4)

[3,4]

The open ball in ℝ is the open interval (3−4, 3+4) = (−1, 7) because distances less than 4 correspond to points inside that interval.

Explanation

The open ball in ℝ is the open interval (3−4, 3+4) = (−1, 7) because distances less than 4 correspond to points inside that interval.

5) Every open ball in a metric space is an open set.

True

False

True, because open balls form a basis for the metric topology; every open ball is itself an open set.

Explanation

True, because open balls form a basis for the metric topology; every open ball is itself an open set.

6) Which metric on ℝ² produces open balls that look like diamonds (rhombuses)?

Euclidean metric d₂

Supremum metric d∞

Manhattan metric d₁

Discrete metric

In the Manhattan (taxicab) metric, the set of points satisfying |x₁−a₁| + |x₂−a₂|

Explanation

In the Manhattan (taxicab) metric, the set of points satisfying |x₁−a₁| + |x₂−a₂|

7) In the discrete metric, every ball of radius less than 1 contains exactly one point.

True

False

True, because d(x,y)=1 for all x≠y. So if r

Explanation

True, because d(x,y)=1 for all x≠y. So if r

8) In (ℝ², d₁), the open ball B((0,0),1) is:

A filled circle

A square with sides of length 2, centered at the origin

A diamond

A hexagon

The Manhattan metric creates points satisfying |x| + |y| < 1, which is a diamond-shaped region.

Explanation

The Manhattan metric creates points satisfying |x| + |y| < 1, which is a diamond-shaped region.

9) If r < s, then B(x,r) ⊂ B(x,s).

True

False

True, because every point within distance r is also within distance s when r

Explanation

True, because every point within distance r is also within distance s when r

10) Which statement correctly describes boundary behavior of an open ball?

It always contains its boundary

It never contains any boundary points

It contains some but not all boundary points

It contains its boundary only if the space is discrete

Open balls never contain boundary points because boundary points satisfy d(x,y)=r, and open balls require d(x,y)

Explanation

Open balls never contain boundary points because boundary points satisfy d(x,y)=r, and open balls require d(x,y)

11) In any metric space, the intersection of two open balls is always an open ball.

True

False

False, because intersections of open balls are open sets but not necessarily balls themselves. They can form irregular shapes.

Explanation

False, because intersections of open balls are open sets but not necessarily balls themselves. They can form irregular shapes.

12) Which metric has open balls that can be the entire space?

Euclidean metric

Discrete metric

Manhattan metric

Supremum metric

In the discrete metric, if r ≥ 1, the open ball around any point includes the entire space.

Explanation

In the discrete metric, if r ≥ 1, the open ball around any point includes the entire space.

13) Two distinct open balls with the same center can never be equal.

True

False

True, because if the radii differ, the balls differ: one is strictly larger. Open balls strictly expand as radius increases.

Explanation

True, because if the radii differ, the balls differ: one is strictly larger. Open balls strictly expand as radius increases.

14) In a metric space (X,d), if B(x,r) = X, which must be true?

The space is bounded

R is greater than or equal to the diameter of the space

X is an isolated point

D is the discrete metric

For the entire space to lie inside the ball, the radius must be at least the diameter of X.

Explanation

For the entire space to lie inside the ball, the radius must be at least the diameter of X.

15) Which describes when two open balls must be disjoint?

When d(x,y) = 0

When d(x,y) > r + s

When d(x,y) < r + s

When r = s

Two balls B(x,r) and B(y,s) are disjoint when their centers are farther apart than the sum of their radii.

Explanation

Two balls B(x,r) and B(y,s) are disjoint when their centers are farther apart than the sum of their radii.