Answer Key — Topology and continuity

Real analysis

Do not share with students before practice tests.

Worksheets — Sets 1–10

| Set | Guidance | | --- | --- | | Set 1 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 2 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 3 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 4 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 5 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 6 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 7 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 8 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 9 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 10 | Topology and continuity — verify written work and accept reasonable drawings. |

Worksheets — Sets 11–20

| Set | Guidance | | --- | --- | | Set 11 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 12 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 13 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 14 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 15 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 16 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 17 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 18 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 19 | Topology and continuity — verify written work and accept reasonable drawings. | | Set 20 | Topology and continuity — verify written work and accept reasonable drawings. |

Review tiers 1–10

• Checklists are parent-judged.
• Written items: verify against practice set patterns.
• Standard text: Topology of R: open and closed sets, compactness, and the Heine–Borel theorem Limits of functions; sequential characterization of limits Continuity; extreme value and intermediate value theorems (with proofs) Uniform continuity and its distinction from pointwise continuity

Practice test scoring

| Tier | Pass guidance | | --- | --- | | 1–3 | Most oral tasks smooth; majority of written correct | | 4–6 | 7/10+ total with clear understanding | | 7–8 | 9/12+ with explanations | | 9–10 | Near-perfect; ready to move on |

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