Results on paths, arcs, first Čech cohomology and dendrites, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Chain lemma — Lemma 2.1. In a connected space covered by open sets, any two points are joined by a finite chain of members with consecutive members meeting.
• Connected open sets are path connected — Theorem 4.1. In a locally connected complete metric space every connected open set is path connected.
• Peano continua — Corollary 4.2. A locally connected continuum and its connected open subsets are path connected; components of open subsets are open.
• Erasing loops — Proposition 2.1. A path between two distinct points can be replaced, inside its image, by a path whose fibres are intervals.
• Paths with interval fibres — Proposition 3.1. The image of a path with interval fibres and distinct ends is an arc between its ends.
• Arc theorem — Theorem 4.1. The image of a path between two distinct points of a metric space contains an arc between them.
• Arcs in connected open sets — Corollary 4.2. Two distinct points of a connected open subset of a locally connected complete metric space are the endpoints of an arc inside it.
• Two arcs — Lemma 5.2. The union of two different arcs with the same endpoints contains a simple closed curve.
• First Čech cohomology — Definition 2.2. The group of classes of nerve cocycles on finite open covers of a compact metric space, with any abelian coefficient group.
• Vanishing criterion — Theorem 3.1. The first Čech cohomology vanishes exactly when every locally constant cocycle on every finite open cover is a system of differences of locally constant functions.
• Two open sets — Corollary 4.1. With vanishing first Čech cohomology, a locally constant function on the intersection of two open sets covering the space is a difference of locally constant functions on the two sets.
• The circle — Corollary 4.3. A simple closed curve has nonzero first Čech cohomology for every nonzero coefficient group.
• Retracts — Corollary 4.4. Vanishing of the first Čech cohomology passes to retracts.
• Extension across a point — Corollary 4.5. With vanishing first Čech cohomology, a locally constant function near a point whose complement is connected extends across the point.
• Classes on closed subsets — Proposition 5.1. In a compact metric space of covering dimension at most one, every first Čech class on a closed subset is the restriction of a class on the space.
• No simple closed curves — Theorem 2.1. A compact metric space of dimension at most one with vanishing first Čech cohomology for a nonzero coefficient group contains no simple closed curve.
• Arcs in a dendrite — Theorem 3.1. Two points of a dendrite are joined by exactly one arc, and the points strictly inside it are exactly the points separating them.
• Cut points — Corollary 3.2. A Peano continuum with more than one point, of dimension at most one and with vanishing first Čech cohomology, has a cut point.

Reading order

Read the lessons in order; the third lesson on Čech cohomology uses none of the first two.