Study Guide — Compactness and connectedness
Introductory topology
Undergraduate study guide for Compactness and connectedness.
What this standard means
- Compactness: open covers and sequential compactness in metric spaces
- Heine–Borel and Bolzano–Weierstrass in R^n
- Connected and path-connected spaces; components
- Compactness of continuous images and extreme value theorem (topological proof)
- Introduction to homotopy and the fundamental group (preview)
_See printable PDF for diagram._
How to use the 20 practice sets
| Sets | When to use | | --- | --- | | 1–5 | Intro — explore together, short written items | | 6–10 | Core skills — diagrams and written practice | | 11–15 | Mixed review — explain thinking | | 16–20 | Stretch — word problems and mastery tasks |
Pacing: 10–15 minutes per session.
How to practice
1. Start from what is given 2. State the logical structure explicitly 3. Check edge cases and quantifiers
_See printable PDF for diagram._
Common mistakes
- Circular reasoning
- Skipping quantifier order
Review and practice tests
1. Start Review 1/10 when sets 1–3 feel comfortable. 2. Move up one review level with little help. 3. Use Practice Test 4/10–6/10 for mid-standard checks. 4. Practice Test 10/10 is the mastery bar for Compactness and connectedness.
- [ ] Writes complete proofs
- [ ] Uses definitions precisely
- [ ] Selects appropriate proof techniques
Materials for this standard
- Practice Problems — 20 printable sets
- Review — 10 difficulty levels
- Practice Test — 10 difficulty levels
- Answer key — for parents and tutors