A topology proof and counterexample workshop

Try

A topology report should identify the space before choosing a theorem. The same underlying set can support usual, discrete, cofinite or other topologies, and a familiar formula can change its continuity behavior when those choices change.

Start with a proposed comparison: usual R versus discrete R. The identity in one direction is continuous and bijective, but not a homeomorphism. A stronger conclusion requires an invariant: discreteness is preserved by every homeomorphism, and only one of these two spaces is discrete.

Predict

Predict why the failure of one proposed map does not prove two spaces are inequivalent. Another map might work. Conversely, a valid homeomorphism needs continuity of its inverse, not merely a plausible one-to-one diagram.

Predict which hypotheses can repair that inverse-continuity gap. Compact source and Hausdorff target suffice for a continuous bijection. The proof passes through closed subsets and compact images, so each condition has a specific role.

Experiment

Use a hypothesis ledger.

Proposed conclusion	Mechanism to verify
Continuity into a product	Every coordinate continuous
Continuity out of a quotient	Continuous representative-independent composite
Inverse continuity	Explicit inverse, open/closed map, or compact-to-Hausdorff theorem
No homeomorphism exists	A proved invariant differs
General convergence criterion	Nets, or a justified first-countability assumption

The ledger is not a substitute for proof. It identifies which definitions and preservation theorems a proof must connect.

Understand

Here is a complete model comparison. Suppose f:X→Y is a continuous bijection with X compact and Y metric. A closed F⊆X is compact. Its image f(F) is compact by continuous-image preservation. Metric Y is Hausdorff, so f(F) is closed. Thus f is a closed map, making its inverse continuous. This argument proves the conclusion without guessing an inverse formula and records exactly where metric structure is used: only to supply Hausdorffness.

For a failed construction, consider the double-origin quotient. The source is the disjoint union of two Hausdorff lines, but the two origin classes cannot be separated after gluing nonzero points. Every quotient neighborhood of an origin contains a small interval of representatives. Two such intervals share an identified nonzero point. This proves non-Hausdorffness directly and shows why source separation does not automatically survive quotienting.

For a product-versus-box audit, the diagonal map t↦(t,t,…) has continuous coordinates and is therefore product-continuous. The box with jth factor (−1/j,1/j) has preimage only zero, so box continuity fails. The exact preimage identifies the broken infinite-intersection step.

A proof portfolio should include a construction, a preservation theorem, a counterexample and an explicit dependency statement. Advanced results such as Tychonoff, Urysohn metrization, Stone–Čech and finite-dimensional embedding have different proof and choice requirements; citing their names without hypotheses is not evidence that an application is valid.

Apply

Choose two spaces and attempt both directions of a proposed map. Record any failed continuity test as a specific open set with nonopen preimage. Then decide whether the failure only rejects that formula or whether an invariant rules out all possible homeomorphisms.

Repair “The finite widget passed, so the general theorem is demonstrated.” Every finite space is compact and every finite T₁ space is discrete. Those facts make finite experiments unable to represent several central counterexamples in this sequence. Use them to inspect definitions, then supply an argument at the intended scope.

Continue

This sequence develops point-set topology through compactification, category and introductory dimension. Independent proof assessment, deeper dimension and embedding proofs, algebraic topology and specialized compactification theory remain separate work. Keep source comparisons honest: covering a syllabus topic with stated dependencies is not the same as completing every proof, examination or project in that source course.