Metric completeness and closure from a basis
| English | 中文 | Pinyin |
|---|---|---|
| complete metric space/kəmˈpliːt ˈmetrɪk speɪs/ | 完备度量空间 | wán bèi dù liàng kōng jiān |
| neighbourhood basis/ˈneɪbəhʊd ˈbeɪsɪs/ | 邻域基 | lín yù jī |
A decision before an answer
- Changing the distance formula can make a sequence Cauchy even when its ordinary values grow without bound. The relevant limit must still be a point of the metric space.
- Your goal: Check a pullback metric using the properties of its defining map.
Read the relationship
- For an injective map h:X→R, d(x,y)=|h(x)−h(y)| is a metric: symmetry and the triangle inequality come from R, and injectivity ensures d(x,y)=0 only when x=y. The map h is an isometry onto its image h(X). If h is not injective, this construction may be only a pseudometric. Bounded distances do not prove a space complete, and a familiar set of points can have very different Cauchy behaviour under different metrics.
- Decide completeness through the image of an isometry.
Which statement holds for d(x,y)=|arctan x−arctan y| on R?
The injective map defines a metric. The image is an open bounded interval, and arctan n tends to a missing endpoint.
Use the defining rule
- The pullback metric is complete exactly when the image h(X) is complete in the ordinary real distance. A closed subset of R is complete. For h(x)=arctan x on R, the image is (−π/2,π/2), which omits its endpoints. The sequence n is Cauchy in the pullback metric because arctan n→π/2, but no real x has arctan x=π/2, so the metric is incomplete. For h(x)=x³, the image is all R and the pullback metric is complete. Topological equivalence alone does not preserve completeness.
- Determine closure using every basic neighbourhood instead of Euclidean intuition.
For the divisor basis in the lesson, which point belongs to closure of {8}?
Closure consists of multiples of 8. The point 16 has every neighbourhood containing 8; U_4 demonstrates why the divisor 4 does not belong.
Check the conditions
- In a topology with a basis, x lies in the closure of A when every basic open neighbourhood of x meets A. This is a membership test, not automatically a Euclidean endpoint operation. The basis must cover the space, and for a point in two basis sets there must be a smaller basis set containing it within their intersection. A point can lie in the closure of a singleton even when it is not the singleton’s own point; that possibility depends on the topology.
- Determine closure using every basic neighbourhood instead of Euclidean intuition.
Under d(x,y)=|arctan x−arctan y|, the integer sequence goes toward a missing image endpoint rather than a point of R, so it is Cauchy without convergence in that space. In the divisor basis, 16 lies in closure of {8}, since any basic set containing 16 also contains 8. But 4 does not: U_4={2,4} is a neighbourhood of 4 missing 8. This explicitly separates multiples from divisors.
For h(x)=x³, d(1,2)=|h(1)−h(2)| is ____.
The transformed coordinates are 1 and 8, with difference 7.
Apply the task format
- On X={2,3,4,…}, take basic sets U_k={n in X: n divides k}, for k≥2. They cover X since x∈U_x; intersections are U_gcd(k,l) when the gcd is at least 2, or empty. A point x lies in closure of {n} exactly when every k divisible by x is also divisible by n: equivalently n divides x. More directly, the smallest basic neighbourhood U_x contains n precisely when n divides x, and every other neighbourhood containing x includes these divisors. Thus closure of {8} consists of positive multiples of 8, not the divisors of 8.
- Determine closure using every basic neighbourhood instead of Euclidean intuition.
A Cauchy limit must belong to the same space. Completeness is a metric property, not just a topological one. For closure, test every neighbourhood of the candidate point, rather than every neighbourhood of the singleton value.
Which answer fits this case?
Check a pullback metric using the properties of its defining map
Two metrics defining the same topology must either both be complete or both be incomplete.
Ordinary distance and the arctan pullback give the usual topology on R, but only ordinary distance is complete.
Keep the distinctions
- complete metric space 完备度量空间 — A metric space in which every Cauchy sequence converges to a point of that space.
- neighbourhood basis 邻域基 — Basic neighbourhoods sufficient to test local topological properties.
- Check a pullback metric using the properties of its defining map.
- Decide completeness through the image of an isometry.
- Determine closure using every basic neighbourhood instead of Euclidean intuition.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.