Real analysis and topology
| English | 中文 | Pinyin |
|---|---|---|
| compact/kəmˈpækt/ | 紧致的 | jǐn zhì de |
| supremum/suːˈpreməm/ | 上确界 | shàng què jiè |
A decision before an answer
- The open interval (0,1) is bounded, but a continuous function on it need not attain a maximum.
- Your goal: Apply sequence and function limit definitions.
Read the relationship
- An epsilon–delta statement controls all sufficiently close inputs. The quantifier order matters.
- Distinguish compactness, connectedness and completeness.
Which subset of the real line is compact?
It is both closed and bounded in the real line.
Use the defining rule
- In real Euclidean space, closed and bounded sets are compact. Do not apply this equivalence to every metric space.
- Use metric-space and elementary topological reasoning.
A convergent real sequence must be:
Convergence gives an eventual bound plus finitely many earlier terms.
Check the conditions
- Continuous images of compact sets are compact, so a real continuous function on a compact domain attains extrema.
- Use metric-space and elementary topological reasoning.
f(x)=x on (0,1) is continuous and bounded but never equals its supremum 1. On [0,1], the same function attains its maximum at 1. The missing endpoint explains why the compact-domain theorem does not apply to the first case.
The supremum of (0,1) is ____.
It is the least upper bound, although it is not in the set.
Apply the task format
- Connectedness rules out a separation into disjoint nonempty open parts. Continuity preserves connectedness; completeness is a separate property.
- Use metric-space and elementary topological reasoning.
A theorem’s conclusion cannot be used before its hypotheses have been checked.
Which answer fits this case?
Apply sequence and function limit definitions
Every bounded continuous function attains its supremum.
f(x)=x on (0,1) does not.
Keep the distinctions
- compact 紧致的 — Every open cover has a finite subcover.
- supremum 上确界 — The least upper bound of a set.
- Apply sequence and function limit definitions.
- Distinguish compactness, connectedness and completeness.
- Use metric-space and elementary topological reasoning.
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.