T.12 · Complétude métrique et fermeture à partir d'une base
GRE Subject Test · GRE Mathematics Sujet 31 2:43 Narration en anglais · Sous-titres anglais + 中文 incrustés
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This unit checks a pullback metric using its defining map, decides completeness through the image of an isometry, and determines closure using every basic neighbourhood instead of Euclidean intuition.
本单元借助定义映射检验拉回度量，通过等距映射的像判断完备性，并用每个基本邻域而不是欧氏直觉来确定闭包。
For an injective map h into R, the distance between h of x and h of y is a metric: symmetry and the triangle inequality come from R, and injectivity makes distance zero only for equal points.
对映到 R 的单射 h，|h(x) − h(y)| 是一个度量：对称性和三角不等式来自 R，单射保证距离为零只发生在点相同时。
h is then an isometry onto its image.
于是 h 是到其像上的等距映射。
The pullback metric is complete exactly when the image is complete.
拉回度量完备，当且仅当像是完备的。
For arctan, the image is the open interval of width pi.
对 arctan，像是宽为 π 的开区间。
The sequence n is Cauchy, because arctan n tends to pi over 2, but no real x has arctan x equal to pi over 2.
数列 n 是柯西列，因为 arctan n 趋于 π/2，但没有实数 x 满足 arctan x = π/2。
The cube map has image all of R, so its metric is complete.
立方映射的像是整个 R，所以它的度量是完备的。
x lies in the closure of A when every basic neighbourhood of x meets A.
当 x 的每个基本邻域都与 A 相交时，x 属于 A 的闭包。
On the integers from 2, take U k as the divisors of k.
在从 2 开始的整数上，取 U_k 为 k 的因数集。
They cover, and intersections are U of the gcd or empty.
它们覆盖全集，交集是 U_gcd 或空集。
The smallest neighbourhood of x is U x, so x is in the closure of n exactly when n divides x.
x 的最小邻域是 U_x，所以 x 属于 {n} 的闭包当且仅当 n 整除 x。
Under the arctan metric the integers head toward a missing image endpoint: Cauchy, without a limit.
在 arctan 度量下，整数趋向像中缺失的端点：是柯西列，但没有极限。
In the divisor basis, 16 lies in the closure of 8, since every basic set containing 16 contains 8.
在因数基下，16 属于 {8} 的闭包，因为包含 16 的每个基本集都包含 8。
But 4 does not: U 4 is 2, 4, missing 8.
但 4 不属于：U₄ = {2, 4}，不含 8。
A Cauchy limit must belong to the same space.
柯西列的极限必须属于同一空间。
Completeness is a metric property, not just a topological one.
完备性是度量性质，不只是拓扑性质。
For closure, test every neighbourhood of the candidate point, not of the singleton's value.
求闭包时，要检验候选点的每个邻域，而不是单点集中那个值的邻域。
Both maps are injective, so both give metrics, each an isometry onto its image.
两个映射都是单射，所以都给出度量，并且都是到其像上的等距映射。
The cube map's image is all of R, so d 1 is complete.
立方映射的像是整个 R，所以 d₁ 完备。
The arctan image is an open interval, so d 2 is incomplete: the sequence n is d 2 Cauchy with no limit.
arctan 的像是开区间，所以 d₂ 不完备：数列 n 在 d₂ 下是柯西列却没有极限。
The smallest basic neighbourhood of x is U x, which meets 6 exactly when 6 divides x, so the closure is the multiples of six.
x 的最小基本邻域是 U_x，它与 {6} 相交当且仅当 6 整除 x，所以闭包是 6 的倍数。
The point six is not interior to its singleton, because U6 also contains two and three: the interior is empty.
点 6 不是其单点集的内点，因为 U₆ 还包含 2 和 3：内部是空集。
Check yourself.
自我检测。
The cube metric is complete; the arctan metric is not.
立方度量完备；arctan 度量不完备。
The closure of 8 is its multiples, and 4 is not among them.
{8} 的闭包是 8 的倍数，4 不在其中。
And the singleton 6 has empty interior.
单点集 {6} 的内部为空。