T.5 · Open sets, compactness and connectedness
GRE Subject Test · GRE Mathematics Topic 14 2:56 English narration · English + 中文 subtitles burned in
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This unit computes closure, interior and boundary in a stated space, applies compactness and connectedness to continuous maps, and distinguishes relative topology from the ambient Euclidean topology.
本单元在指定空间中计算闭包、内部和边界，把紧性和连通性应用于连续映射，并区分相对拓扑与外围欧氏拓扑。
An open set contains a small ball around each of its points.
开集包含其每个点周围的一个小球。
The interior is the set of such points, the closure adds all limit points, and the boundary is closure minus interior.
内部是这样的点的集合，闭包加上所有极限点，边界是闭包减去内部。
For 0 to 1, half-open, in R: interior 0, 1 open, closure 0, 1 closed, boundary the two endpoints.
对 R 中的 ，边界是两个端点。
In a subspace X, an open set is X intersected with an ambient open set.
在子空间 X 中，开集是 X 与外围开集的交。
So 0 to 1, half-open, is open relative to 0, 2, by intersecting with minus 1, 1, although it is not open in R.
所以 是开的（与 (−1, 1) 相交得到），尽管它在 R 中不是开集。
Always state which space defines neighbourhoods.
一定要说明是哪个空间定义邻域。
Compactness means every open cover has a finite subcover; in R n that is closed and bounded.
紧性指每个开覆盖都有有限子覆盖；在 Rⁿ 中就是有界闭集。
A continuous image of a compact set is compact, so extrema are attained.
紧集的连续像是紧的，所以能取到最值。
Connected subsets of R are the intervals, and a continuous image of a connected set is connected: the intermediate value theorem.
R 的连通子集就是区间，连通集的连续像是连通的：这就是介值定理。
Let X be 0, 1, and U equals 0 to a half. U is X intersected with an open interval, so U is open in X; its closure in X is 0 to a half, closed, and its boundary in X is a half alone.
设 X = ，U = ，在 X 中的边界只有 1/2。
In the real line the boundary is 0 and a half.
在实数轴上，边界是 0 和 1/2。
Zero is interior relative to X.
0 相对于 X 是内点。
Heine-Borel's closed-and-bounded test is a Euclidean-space theorem.
海涅–博雷尔的“有界闭”判别法是欧氏空间中的定理。
Connectedness and compactness answer different questions.
连通性和紧性回答的是不同的问题。
A is X intersected with minus 1, 1, so it is open in X and its relative interior is A itself.
A = X ∩ (−1, 1)，所以它在 X 中是开的，相对内部就是 A 本身。
Its closure is 0, 1, closed, with boundary 1.
它的闭包是 ，边界是 {1}。
It is not closed in X: 1 minus 1 over n tends to the missing point 1.
它在 X 中不是闭的：1 − 1/n 趋向缺失的点 1。
In the real line the interior and boundary would differ.
在实数轴上，内部和边界会不同。
The finite set 0, 2 is compact but disconnected.
有限集 {0, 2} 是紧的，但不连通。
The interval 0, 1, open, is connected but not compact: the identity function there attains neither a maximum nor a minimum.
开区间 (0, 1) 连通但不紧：其上的恒等函数既取不到最大值也取不到最小值。
Connectedness supplies intermediate values; compactness supplies extremes.
连通性提供介值；紧性提供最值。
Neither substitutes for the other.
两者不能相互替代。
Check yourself.
自我检测。
The boundary of 0 to 1 in R is the two endpoints.
是开的。 紧性给出能取到的最值；连通性给出介值。
0 to 1 is open relative to 0, 2. Compactness gives attained extremes; connectedness gives intermediate values. And the set 0, 2 is not connected.
集合 {0, 2} 不连通。