Milestone map
Milestone map
2 milestones
Do the problem sets and write them up
8–15 weeks (14 hrs/week)
Take Topology Without Tears for free and do the exercises on topological spaces, bases, continuity, connectedness, compactness and metric spaces. Write every solution up in LaTeX and keep the source and the PDF in one public repo as you go.
Proof required
Link your public repo with the LaTeX source and the PDF of the exercises on topological spaces, bases, continuity, connectedness, compactness and metric spaces, and show your self-check (Self-check without an answer key).
What gets checked
- Every proof is complete and correct.
- Counterexamples really fail the property named.
- Each step names the definition it uses.
Common mistakes
- Copying posted solutions: verifiers read the reasoning and the history.
- A proof that skips the hard step.
- Asking for review before the self-check.
Powstik Guide
A public repo with your LaTeX write-ups of the exercises on topological spaces, bases, continuity, connectedness, compactness and metric spaces.
Steps
- Make a public repo for this course.
- Do one problem set at a time; typeset it in LaTeX and commit.
- Self-check: The book has no answer key. For each problem, name the definition or theorem every step uses; a step you cannot name is a gap. Then swap write-ups with a study partner before you ask for review.
- Fix what the self-check shows, and say what you fixed.
- Ask three people who know the topic to check it.
Template
README Course: Topology Without Tears (Topology Without Tears by Sidney Morris (free book)) What is here (problem sets, PDF): Self-check result: The hardest proof and why:
What gets sent back
- Answers with no reasoning.
- Photos of handwriting instead of typeset work.
- Self-check not shown.
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Every proof is complete and correct.
- Counterexamples really fail the property named.
- Each step names the definition it uses.
- Pick two proofs and check every step: each argument must be correct and complete, not just present.
- Ask about one step; the answer should be clear.
You'll sign in first, then come straight back here.
Write a topology proof set
1–3 weeks (5–8 hrs/week)
Use what you learned. Make a proof set of 8 results that compares two topologies on the real line (for example, the usual one and the lower-limit one): which sets are open, compact and connected, with proofs. Typeset it in LaTeX (or a notebook with LaTeX maths) in your public repo.
Proof required
Link a proof set of 8 results that compares two topologies on the real line (for example, the usual one and the lower-limit one): which sets are open, compact and connected, with proofs, in your public repo.
What gets checked
- Each claim is proved or disproved with a counterexample.
- Every argument is complete.
- The comparison table matches the proofs.
Common mistakes
- A gap in the main argument.
- A result stated but not proved.
Powstik Guide
A proof set of 8 results that compares two topologies on the real line (for example, the usual one and the lower-limit one): which sets are open, compact and connected, with proofs.
Steps
- Write one line: what you will prove or show.
- Draft it, then read it cold the next day.
- Ask someone to read it and mark the first step they doubt.
- Fix it and typeset the final version.
- Ask three people to check it.
Template
Title: What this shows: Main result (statement): Proof: What I would look at next:
What gets sent back
- A main step left unproved.
- Not clearly the person's own work.
- Claims more than it proves.
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Each claim is proved or disproved with a counterexample.
- Every argument is complete.
- The comparison table matches the proofs.
- Every argument must be correct and complete; mark the first gap you find.
- Check that the work is the person's own.
You'll sign in first, then come straight back here.