به انگلیسی

Distance as a short list of rules

Mathematics · Topology · ages 20-21

Name ______________________ Date ____________

Use the discrete metric on the reals. For any real center a, how many points belong to the open ball B(a,1)?
Answer: ______________
On the reals, propose d(x,y)=|x-y|^2. For x=0, y=1 and z=2, which metric axiom fails?
- Nonnegativity of every proposed distance
- Symmetry when the two points exchange roles
- The triangle inequality
- Zero distance exactly between equal points
Which proposed condition is NOT required of every metric?
- d(x,y)>=0 for every allowed pair x,y
- d(x,z)>=d(x,y)+d(y,z)
- d(x,y)=d(y,x) for every allowed pair x,y
Use the usual real metric d(x,y)=|x-y|. For x_n=1/n with n=1,2,3,..., how many terms lie outside the open ball B(0,0.01)?
Answer: ______________
For f(x)=5x with usual real input and output metrics, what is the largest delta such that |x-a|<delta guarantees |f(x)-f(a)|<0.3 for every real x?
Answer: ______________
For a function between metric spaces, continuity at a is equivalent to this condition: every sequence x_n tending to a has f(x_n) tending to f(a).
Circle one: True False
On C([0,2]), use d(f,g)=max |f(t)-g(t)| over [0,2]. For f(t)=3t and g(t)=t, what is d(f,g)?
- 4, the maximum of 2t on [0,2]
- 2, the gap evaluated only at the midpoint t=1
- 6, the maximum of f without subtracting g
- 0, the gap evaluated only at the left endpoint t=0
Every function from a discrete metric space to any metric space is continuous at every point.
Circle one: True False
Suppose d(x_n,x)=1/n^2 for positive integers n. What is the smallest positive integer N such that d(x_n,x)<0.01 for every n>=N?
Answer: ______________
Let d be a metric on X and define e(x,y)=2d(x,y). How do convergence and limits compare?
- Every d-convergent sequence loses its limit when its distances are doubled
- A sequence converges under e only when it eventually equals its limit
- Each d-limit changes to another point because all distances are doubled
- Both metrics have the same convergent sequences and limits

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

1 · Only a has distance less than 1 from a. Every other point has distance exactly 1 and is excluded from this open ball.
The triangle inequality · The direct distance is 4, while the two distances through 1 sum to 2. The triangle inequality would require 4<=2.
d(x,z)>=d(x,y)+d(y,z) · The triangle axiom uses <=. For the usual real metric, x=z=0 and y=1 give direct distance 0 and a two-distance sum of 2, disproving the proposed >= condition.
100 · Outside this open ball means 1/n>=0.01. Precisely the positive indices 1 through 100 satisfy that condition, including the boundary term.
0.06 · The output gap is 5|x-a|, so delta=0.3/5=0.06 works. Any larger delta admits input gap 0.06, giving output gap exactly 0.3.
True · Continuity controls every approaching sequence. Failure of continuity gives approaching inputs whose output distances stay at least a fixed positive tolerance.
4, the maximum of 2t on [0,2] · The absolute gap is |3t-t|=2t on [0,2]. Its maximum occurs at t=2 and equals 4.
True · At any input a, radius delta=1 admits only a. The output distance is therefore zero, meeting every positive output tolerance.
11 · The strict inequality requires n^2>100, hence n>10. Thus the smallest tail index is 11; n=10 gives equality and does not qualify.
Both metrics have the same convergent sequences and limits · The scaled distance satisfies all metric axioms. For every proposed limit x, e(x_n,x)=2d(x_n,x) tends to zero exactly when d(x_n,x) does.