Distance as a short list of rules

Mathematics · Topology · ages 20-21

Name ______________________ Date ____________

In the reals with the usual metric, consider x_n equal one over n. How many terms lie outside the ball of radius zero point zero one about zero?
Answer: ______________
Which statement is NOT a metric axiom?
- d(x, y) equals d(y, x)
- d(x, y) is 0 exactly when x equals y
- d(x, z) is at least d(x, y) plus d(y, z)
- d(x, z) is at most d(x, y) plus d(y, z)
Which statement is NOT a metric axiom?
- Distances are never negative
- Distances satisfy the reversed triangle inequality
- Distance is symmetric in its two points
In the reals with the usual metric, look at 1 over n. How many terms lie outside the ball of radius 0.01 about zero?
Answer: ______________
Suppose d(x_n, x) equals one over n squared. What is the distance at n equal ten?
Answer: ______________
In metric spaces, f is continuous at a exactly when approaching sequences map to approaching values.
Circle one: True False
On continuous functions, d(f, g) is the maximum of |f minus g|. Why does the triangle inequality hold?
- The pointwise triangle inequality passes to the maximum
- All continuous functions are bounded by one
- The maximum of a sum equals the sum of the maxima
- Continuity forces all distances to be zero
In metric spaces, f is continuous at a exactly when x_n tending to a forces f(x_n) tending to f(a). True or false?
Circle one: True False
Suppose d of x_n and x equals 1 over n squared, giving distance 0.01 at n equal 10. What does that say about convergence?
Answer: ______________
Two metrics on one set disagree about which sequences converge. What does this show?
- One of the two cannot be a metric
- The underlying set must be finite
- Limits are never unique in metric spaces
- Convergence is a property of the metric, not of the bare set

Answer key

For grown-ups. Fold this page away before handing over the rest.

Distance as a short list of rules W1-mt_A3TMb8wL6k-s1

100 · One over n reaches zero point zero one at n equal one hundred, so the first one hundred terms sit outside.
d(x, z) is at least d(x, y) plus d(y, z) · The triangle inequality caps distance by the detour sum. The reversed inequality is false in general.
Distances satisfy the reversed triangle inequality · The triangle inequality caps distance by the detour sum, never the reverse.
100 · 1 over n reaches 0.01 at n equal 100, so the first 100 terms sit outside.
0.01 · One over one hundred is zero point zero one.
True · Sequential continuity matches ball continuity in metric spaces.
The pointwise triangle inequality passes to the maximum · Pointwise control at every point survives taking the maximum over points.
True · True. Sequential continuity matches epsilon ball continuity in metric spaces.
0.01 · Distances shrink to 0, which is convergence to x.
Convergence is a property of the metric, not of the bare set · The same points with different rulers give different arrivals. Measure matters.