Continua, arcs and cut points

Locally connected continua with complete proofs: connected open sets are path connected, every path contains an arc, first Čech cohomology of a compact metric space by finite covers, and the theorem that a Peano continuum of dimension at most one with vanishing first Čech cohomology is a dendrite and has cut points, which is the last step of OpenAI's proof (2026) of the Singer conjecture in dimension four outside the middle degree.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Paths in locally connected spaces
2. Every path contains an arc
3. First Čech cohomology of a compact metric space
4. Dendrites and cut points

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Čech cohomology · Cohomology of quasi-coherent sheaves
• Covering dimension and finite trivializing covers · Index theory of elliptic operators (Index theory of elliptic operators — Hermite and Schwartz tools)
• Point-set topology (core course C90) and the definition of covering dimension from the lesson on covering dimension and finite trivializing covers; the course proves the other facts it uses.