기저로부터의 메트릭 완전성과 폐쇄성
도입됨| English | 의미 |
|---|---|
| complete metric space/kəmˈpliːt ˈmetrɪk speɪs/ | 완비 계량 공간 |
| neighbourhood basis/ˈneɪbəhʊd ˈbeɪsɪs/ | 인접 기저 |
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?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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}?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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 ____.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
정의 map의 성질을 이용하여 당김 계량(pullback metric) 검증
Two metrics defining the same topology must either both be complete or both be incomplete.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.