Open Balls: Comprehensive Assessment Quiz

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1) In any metric space (X,d), an open ball B(x,r) is always a subset of X.

True

False

True, because B(x,r) is defined using distance within X, so all points in the ball must belong to X by definition.

Explanation

True, because B(x,r) is defined using distance within X, so all points in the ball must belong to X by definition.

About This Quiz

Open Balls: Comprehensive Assessment Quiz - Quiz

This quiz brings everything together for a full, comprehensive understanding of open balls. You’ll apply your knowledge to various metrics, examine how open balls define the topology of a space, and explore subtle behaviors that differ across Euclidean, supremum, Manhattan, and discrete metrics. You’ll identify which sets qualify as open... see moreballs, understand how radii interact with membership, and analyze unusual cases where open balls behave differently than expected. From simple one-dimensional intervals to geometric regions in ℝ², this test ensures you can recognize and reason about open balls in any metric context. By the end, you’ll not only know the definitions — you’ll be able to interpret shapes, compare metrics, describe topological properties, and evaluate how open balls form the backbone of metric space theory. This is the ultimate reinforcement quiz for mastering the concept!
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2) What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) Which of the following describes the open ball B(x,r) correctly?

All points whose distance from x is less than or equal to r

All points whose distance from x is exactly r

All points whose distance from x is strictly less than r

All points whose distance from x is greater than r

An open ball includes points y such that d(x,y)

Explanation

An open ball includes points y such that d(x,y)

3) For any metric d, the open ball B(x,0) contains exactly one point.

True

False

True, because d(x,y)

Explanation

True, because d(x,y)

4) In the metric space (ℝ,d) with d(x,y)=|x−y|, the set B(2,3) is:

(2−3, 2+3)

[2−3, 2+3]

(2,5)

[2,5)

In a metric space, the open ball B(x,r) centered at point x with radius r consists of all points that are less than r units away from x. So, B(2,3) represents all points y such that |y - 2| < 3. This results in the interval (2 - 3, 2 + 3), which simplifies to the interval (-1, 5). However, since we are considering only the boundaries that do not include the endpoints in an open ball, the correct representation is (2 - 3, 2 + 3) which is option A.

Explanation

5) Which of the following sets are open balls in (ℝ²,d₂) where d₂ is the Euclidean metric?

A perfect filled circle (including boundary)

A filled circle without boundary

A square centered at a point

An ellipse defined via Euclidean distance

Euclidean open balls are exactly open disks—circles without their boundary.

Explanation

Euclidean open balls are exactly open disks—circles without their boundary.

6) In any metric space, every open ball is an open set.

True

False

True, because metric topologies are generated by open balls, and each open ball is open by definition.

Explanation

True, because metric topologies are generated by open balls, and each open ball is open by definition.

7) Consider d(x,y)=|x−y|² on ℝ. What is the shape of the open ball B(0,1)?

(−1,1)

(−1,1) \ {0}

(−1,1) but with a modified radius criterion

(−1,1) does not describe it correctly

d(x,0)=|x|²

Explanation

d(x,0)=|x|²

8) In the discrete metric d(x,y)=1 if x≠y and 0 otherwise, which of the following are true? Select all that apply.

B(x,1) = {x}

B(x,2) = X

Every subset of X is open

Open balls may contain more than one point

B(x,1) contains only x since d(x,y) < 1 forces y = x. If r ≥ 1, B(x,r) = X. Every subset is open because all points are isolated.

Explanation

B(x,1) contains only x since d(x,y) < 1 forces y = x. If r ≥ 1, B(x,r) = X. Every subset is open because all points are isolated.

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9) Which metric produces diamond-shaped open balls in ℝ²?

D₂

D₁

D∞

Discrete metric

The Manhattan (L¹) metric produces diamond-shaped open balls.

Explanation

The Manhattan (L¹) metric produces diamond-shaped open balls.

10) If two metrics d₁ and d₂ generate the same collection of open balls, they induce the same topology.

True

False

True, because open balls generate the topology, so identical collections of open balls imply identical open sets.

Explanation

True, because open balls generate the topology, so identical collections of open balls imply identical open sets.

11) In any metric space, open balls determine the metric uniquely.

True

False

False, because different metrics can generate the same open balls and hence the same topology (e.g., equivalent metrics).

Explanation

False, because different metrics can generate the same open balls and hence the same topology (e.g., equivalent metrics).

12) Which statements about open balls in any metric space are always true? Select all that apply.

Open balls form a basis for the topology

For any x, B(x,r) ⊆ B(x,s) whenever r < s

The union of two open balls is an open ball

The intersection of two open balls may not be an open ball

Open balls form a basis, smaller balls fit inside larger ones, and intersections of open balls may not be balls themselves. Unions of balls are open but not necessarily balls.

Explanation

Open balls form a basis, smaller balls fit inside larger ones, and intersections of open balls may not be balls themselves. Unions of balls are open but not necessarily balls.

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13) In any metric space, every point inside an open ball is the center of another open ball fully contained in the original one.

True

False

True, because if y ∈ B(x,r), you can take ε = r − d(x,y) > 0 so that B(y,ε) ⊂ B(x,r).

Explanation

True, because if y ∈ B(x,r), you can take ε = r − d(x,y) > 0 so that B(y,ε) ⊂ B(x,r).

14) In a metric space (X,d), which of the following must hold for every open ball B(x,r)? Select all that apply.

It is nonempty

It contains its center

Its complement is closed

It is equal to the intersection of all open sets containing the center

Every open ball contains x and is therefore nonempty. Its complement need not be closed in all metric spaces, and intersections of all open neighborhoods define closures, not balls.

Explanation

Every open ball contains x and is therefore nonempty. Its complement need not be closed in all metric spaces, and intersections of all open neighborhoods define closures, not balls.

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