Every connected component of an open subset of ${\mathbb{R}}^n$ is open and polygonally connected

Depends on

• For an open subset of $\mathbb{R}^n$, connectedness, path-connectedness and polygonal connectedness are equivalent
• Open ball, closed ball and sphere in a metric space
• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
• A finite concatenation of straight segments in $\mathbb{R}^n$ is a continuous path
• Every path-connected space is connected, and every path component lies inside a component
• Connected components, quasicomponents, and totally disconnected spaces

Used by

• The index of a cycle is locally constant off its trace and vanishes far from it Corollary
• Conformal removability of compact sets Definition
• The Neumann kernel is spanned by the componentwise constants Example
• A C² leaf meets a local box transversal in at most countably many points Lemma
• A conforming averaging companion for Crouzeix–Raviart functions Lemma
• A finitely cornered regular plane curve separates without choice Lemma
• A positive-height diagram has a defect region Lemma
• An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K Lemma
• Analytic-boundary exhaustion of a plane domain Lemma
• Arc complements and accessible Jordan boundary points Lemma
• Every plane triangulation with at least three vertices is connected Lemma
• Finite planar graph disk cuts and Euler count Lemma
• Finite plane graph ear and face facts Lemma
• Heat-ball chains reach earlier points Lemma
• Jordan–Schönflies extension for plane curves Lemma
• Maximum principle for a compact logarithmic potential Lemma
• Positive transverse accessibility is a preorder Lemma
• Riemann maps of Jordan domains extend to homeomorphisms of the closures Lemma
• The broken comparison expression is a norm Lemma
• The broken gradient is a norm on the homogeneous Crouzeix–Raviart space Lemma
• The complement of a polygonal arc in ℝ² is polygonally connected Lemma
• Two disjoint circles in the two-sphere cobound an annulus Lemma
• A two-connected plane graph of order at least three is maximal exactly when every face is triangular Proposition
• Compactly supported dbar solutions on complex Euclidean space Theorem
• Energy uniqueness for homogeneous Dirichlet waves on bounded domains Theorem
• Local Riesz decomposition of a plane subharmonic function Theorem
• Strong parabolic maximum principle Theorem
• The winding number is constant on each connected component of the complement of the trace Theorem
• Weak Neumann solvability on the mean-zero subspace Theorem
• Zero weak gradient gives componentwise constants Theorem

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Locally connected space (standard reference, not scraped)