Tập mở, tính compact và tính liên thông
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| interior/ɪnˈtɪərɪə/ | The points of a set that have a whole open ball inside the set. |
| closure/ˈkləʊʒə/ | A set together with all of its limit points. |
| connected/kəˈnektɪd/ | Impossible to split into two disjoint nonempty open pieces. |
Open depends on where you stand
- $[0, 1)$ is not open in $\mathbb R$: no ball around $0$ fits inside. But inside the space $X = [0, 2]$, it is $X \cap (-1, 1)$ — open in $X$.
- GRE topology tests interior 内部, closure 闭包 and boundary, relative topology, compactness and connectedness.
Interior, closure, boundary
- Interior: points with a ball inside the set. Closure: the set plus its limit points. Boundary: closure minus interior.
- In $\mathbb R$: $[0,1)$ has interior $(0,1)$, closure $[0,1]$, boundary $\{0, 1\}$. Open and closed are not opposites: $\varnothing$ and the whole space are both.
The closure of $\mathbb Q$ in $\mathbb R$ is
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Every real number is a limit of rationals.
The boundary of $(0,1) \cup (1,2)$ in $\mathbb R$ is
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Closure [0, 2] minus interior (0,1) ∪ (1,2).
Relative topology
- In a subspace $X$, the open sets are $X \cap U$ with $U$ open in the ambient space.
$X = [0,1]$, $U = [0, \tfrac12) = X \cap (-1, \tfrac12)$: open in $X$. In $X$ its closure is $[0, \tfrac12]$ and its boundary is $\{\tfrac12\}$ — 0 is an interior point relative to $X$.
![Number line showing U = 0, 1/2) inside X = [0, 1]: boundary {1/2} in X versus {0, 1/2} in ℝ
![0, 1/2) inside X = [0, 1]: boundary {1/2} in X versus {0, 1/2} in ℝ](/learn-media/gre_mathematics/assets/1401.svg){kind=link}
In $X = [0, 2]$, the set $[0,1)$ is open because it equals $X$ intersected with an ____ subset of $\mathbb R$.
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For example (−1, 1).
Compactness
- Every open cover has a finite subcover. In $\mathbb R^n$: compact $\iff$ closed and bounded. In metric spaces: compact $\iff$ sequentially compact $\iff$ complete and totally bounded.
- Continuous images of compact sets are compact, so continuous real functions attain extrema there.
Connectedness
- Connected 连通的: no split into two disjoint nonempty relatively open parts. In $\mathbb R$, the connected sets are exactly the intervals.
- Continuous images of connected sets are connected (the IVT). Path-connected implies connected; the converse can fail (the topologist's sine curve).
Compactness and connectedness are independent: $\{0, 1\}$ is compact but not connected; $(0, 1)$ is connected but not compact.
Compact, connected, both or neither?
Separate compactness from connectedness for subsets of ℝ.
Which subsets of $\mathbb R$ are connected?
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The connected subsets of ℝ are intervals, including single points.
Every compact subset of $\mathbb R$ is connected.
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False. {0, 1} is closed and bounded, hence compact, but it is not connected: the connected subsets of ℝ are intervals.
Recap
- Name the ambient space before calling a set open or closed.
- Boundary = closure − interior.
- Heine–Borel is for $\mathbb R^n$; intervals are the connected subsets of $\mathbb R$.
- interior: The points of a set that have a whole open ball inside the set.
- closure: A set together with all of its limit points.
- connected: Impossible to split into two disjoint nonempty open pieces.