RefacteThe mechanism
Make the remainder carry the obstruction
A proposed countable partition into closed sets is tested by looking at what remains after its pieces are removed. Closedness in the real line is essential. The argument uses a nonempty closed remainder and Baire category to expose the contradiction.
What becomes elementary: The global partition question is concentrated into a topological obstruction on a carefully chosen remainder.
Keep ambient closedness distinct from relative closedness, and check nonemptiness before applying Baire category.
For another construction, explore the mechanism calculi. Edition dates, sources and reference chapters identify the scope of each study.
The distinction between being closed in the real line and relatively closed in an interval matters here. The note develops the obstruction using point-set topology and Baire category.
Background
Closed sets and the Baire category theorem.
Read the full exposition
The complete argument is available as a PDF.
Revision history ·
Revised September 22, 2026: explicitly established that the closed remainder is nonempty before applying the Baire category theorem.