Open sets, compactness and connectedness
| English | 中文 | Pinyin |
|---|---|---|
| connectedness/kəˈnektɪdnəs/ | 连通性 | lián tōng xìng |
| relative topology/ˈrelətɪv təˈpɒlədʒi/ | 相对拓扑 | xiāng duì tuò pū |
A decision before an answer
- The interval [0,1) is open in one natural space and neither open nor closed in another. The surrounding space is part of the question.
- Your goal: Compute closure, interior and boundary in a stated space.
Read the relationship
- In a metric space, an open set contains a small ball around each of its points. The interior consists of such points; the closure includes all limit points; the boundary is closure minus interior. Open and closed are not mutually exclusive labels; the empty set and the whole space are both. In R, the set [0,1) has interior (0,1), closure [0,1] and boundary {0,1}.
- Apply compactness and connectedness to continuous maps.
What is the boundary of (0,1) as a subset of R?
The closure is [0,1] and the interior is (0,1); removing the latter leaves both endpoints.
Use the defining rule
- In a subspace X, an open set has the form X intersected with an ambient open set. Thus [0,1) is open relative to [0,2], using intersection with (-1,1), although it is not open in R. A set may also be relatively closed without being closed in the ambient space. Always state which space defines neighbourhoods and which metric is used.
- Distinguish relative topology from the ambient Euclidean topology.
Which subset of R is connected but not compact?
An open interval is connected but not closed in R, so it is not compact. The other options are compact or disconnected.
Check the conditions
- Compactness means every open cover has a finite subcover. In Euclidean R^n, Heine–Borel makes this equivalent to closed and bounded. In a metric space, compactness is equivalent to sequential compactness; completeness and boundedness alone do not suffice in arbitrary metric spaces. A continuous image of a compact set is compact, giving attained maxima and minima for real continuous functions on a nonempty compact domain.
- Distinguish relative topology from the ambient Euclidean topology.
Let X=[0,1] with its usual relative topology. The set U=[0,0.5) equals X intersected with (-1,0.5), so U is open in X. Its closure in X is [0,0.5], and its boundary in X is {0.5}; zero is an interior point relative to X. For a continuous f on X with f(0)<0<f(1), connectedness ensures a zero, while compactness separately ensures attained extrema.
The number of boundary points of [0,0.5) relative to X=[0,1] is ____.
Only 0.5 is a relative boundary point. Zero has a relative neighbourhood contained in the set.
Apply the task format
- Connected sets cannot be separated into two disjoint nonempty relatively open parts; connected subsets of R are precisely intervals. A continuous image of a connected set is connected, which yields the intermediate value theorem. Path connectedness implies connectedness, but not conversely in every space. A compact set need not be connected, and a connected set need not be compact; a finite two-point set and an open interval supply the contrasting cases.
- Distinguish relative topology from the ambient Euclidean topology.
Heine–Borel's closed-and-bounded test is a Euclidean-space theorem. Connectedness and compactness answer different questions.
Which answer fits this case?
Compute closure, interior and boundary in a stated space
Every continuous function on a bounded open interval attains a maximum.
f(x)=x on (0,1) has supremum 1 but no maximum. Compactness of the domain would supply the missing hypothesis.
Keep the distinctions
- relative topology 相对拓扑 — Open subsets inherited by intersecting a subspace with ambient open sets.
- connectedness 连通性 — Absence of a separation into disjoint nonempty relatively open subsets.
- Compute closure, interior and boundary in a stated space.
- Apply compactness and connectedness to continuous maps.
- Distinguish relative topology from the ambient Euclidean topology.
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.