Open Balls: Reasoning & Proof Logic Quiz

1) To prove that an open ball B(x,r) is an open set, it is enough to show that every point in the ball is the center of a smaller ball contained in it.

True

False

True, because if every point y in B(x,r) has an ε-ball around it fully contained in B(x,r), then B(x,r) satisfies the definition of an open set.

Explanation

True, because if every point y in B(x,r) has an ε-ball around it fully contained in B(x,r), then B(x,r) satisfies the definition of an open set.

2) Which logical step completes the proof that open balls are open? Suppose y ∈ B(x,r). Define ε = r − d(x,y). Then:

B(x,r) ⊆ B(y,ε)

B(y,ε) ⊆ B(x,r)

B(x,r) = B(y,ε)

B(x,r) ∩ B(y,ε) = ∅

Because ε = r − d(x,y) > 0, any point z with d(y,z)

Explanation

Because ε = r − d(x,y) > 0, any point z with d(y,z)

3) Showing that a point y is in an open ball B(x,r) requires demonstrating that d(x,y) < r.

True

False

True, because this is exactly the definition of membership in an open ball: the distance from the center must be strictly less than the radius.

Explanation

True, because this is exactly the definition of membership in an open ball: the distance from the center must be strictly less than the radius.

4) To prove two open balls B(x,r) and B(y,s) intersect, a sufficient approach is to show:

The centers are equal

D(x,y) < r + s

(x,y) = r + s

R = s

If the distance between centers is less than the sum of radii, then the balls intersect; this follows from basic geometric reasoning and the triangle inequality.

Explanation

If the distance between centers is less than the sum of radii, then the balls intersect; this follows from basic geometric reasoning and the triangle inequality.

5) Which reasoning steps are necessary to prove that if B(x,r) = B(y,r) for some r > 0, then x = y? Select all that apply.

Assume x ≠ y and derive a contradiction

Use the fact that x ∈ B(y,r)

Use symmetry of the metric

Use that d(x,y) = 0 implies x = y

If the balls are equal, then x is in B(y,r), so d(x,y) < r. Similarly, y is in B(x,r). Combining these and assuming x ≠ y leads to a contradiction unless d(x,y) = 0, which forces x = y.

Explanation

If the balls are equal, then x is in B(y,r), so d(x,y) < r. Similarly, y is in B(x,r). Combining these and assuming x ≠ y leads to a contradiction unless d(x,y) = 0, which forces x = y.

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6) To show a set U is open using open balls, which proof strategy is correct?

Show U contains at least one open ball

Show every point of U is contained in an open ball fully within U

Show U is the union of two closed balls

Show U contains its boundary

A set is open if every point has an open ball around it contained entirely in that set.

Explanation

A set is open if every point has an open ball around it contained entirely in that set.

7) To prove that open balls form a basis for a metric topology, one must show that every open set can be expressed as a union of open balls.

True

False

True, because the basis definition requires that every open set be representable as a union of basis elements—in this case, open balls.

Explanation

True, because the basis definition requires that every open set be representable as a union of basis elements—in this case, open balls.

8) Which reasoning step completes the proof that a metric is uniquely determined by its collection of open balls with radii 1/n?

Use the reverse triangle inequality

Show that distances can be approximated from inside by shrinking balls

Show that every closed ball is the intersection of open balls

Use the completeness of the metric space

The idea is that metrics are determined by how small balls behave; the radii 1/n allow one to approximate exact distances through nesting balls.

Explanation

The idea is that metrics are determined by how small balls behave; the radii 1/n allow one to approximate exact distances through nesting balls.

9) To prove the triangle inequality using ball containment logic, which steps are valid? Select all that apply.

Show y ∈ B(x,r) and z ∈ B(y,s) implies z ∈ B(x, r+s)

Use contradiction by assuming d(x,z) ≥ r + s

Use properties of open sets to deduce point membership

Convert point membership inequalities to metric inequalities

Containment arguments translate into inequalities. Showing z ∈ B(x,r+s) is equivalent to proving d(x,z) < r+s, which connects directly to the triangle inequality.

Explanation

Containment arguments translate into inequalities. Showing z ∈ B(x,r+s) is equivalent to proving d(x,z) < r+s, which connects directly to the triangle inequality.

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10) You want to prove that in a metric space, if r < s, then B(x,r) ⊆ B(x,s). What is the most concise reasoning step?

A smaller radius implies a stronger inequality for membership

Triangle inequality ensures inclusion

Open balls are convex

All metrics are monotone

Membership in B(x,r) requires d(x,y)

Explanation

Membership in B(x,r) requires d(x,y)

11) To show two metrics generate the same open sets, it is sufficient to show that every open ball in one metric contains an open ball of the other metric around the same center.

True

False

True, because this ensures that open sets in one metric can be built by unions of open balls from the other, implying topological equivalence.

Explanation

True, because this ensures that open sets in one metric can be built by unions of open balls from the other, implying topological equivalence.

12) To show that a sequence in an open ball eventually remains inside a smaller open ball, which reasoning tools are typically used?

Triangle inequality

Convergence definition

Existence of an ε-ball around the limit point

Continuity of the metric

Convergence ensures points get close to the center, the ε-ball shows containment, and the triangle inequality helps bound distances.

Explanation

Convergence ensures points get close to the center, the ε-ball shows containment, and the triangle inequality helps bound distances.

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13) Suppose x ≠ y. To show there exist disjoint open balls around x and y, use:

Inequality d(x,y) > 0

Radii r = s = d(x,y)

Radii r = 0

If x ≠ y, then d(x,y) > 0, and radii less than half that distance ensure the balls remain disjoint.

Explanation

If x ≠ y, then d(x,y) > 0, and radii less than half that distance ensure the balls remain disjoint.

14) The proof that open balls are open relies fundamentally on the triangle inequality.

True

False

True, because the triangle inequality provides the key inequality that ensures small balls around a point remain inside the larger ball.

Explanation

True, because the triangle inequality provides the key inequality that ensures small balls around a point remain inside the larger ball.

15) In proving that a metric space topology is Hausdorff, which ball-based arguments are valid? Select all that apply.

If x ≠ y, choose disjoint balls around them

Use radii smaller than half the distance between the points

Use that open balls shrink to the center as radius → 0

Use completeness of the metric space

Disjoint balls around distinct points show the Hausdorff property. Completeness is unrelated.

Explanation

Disjoint balls around distinct points show the Hausdorff property. Completeness is unrelated.

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