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.