MathDB

Let G be a domain (connected open set) in the

R^2

plane whose boundary is locally connected. Prove that for every point q of the boundary of G there exists a simple arc

v_q

in which

q\in v_q

and

v_q\setminus\{q\}\subset G

.
other questions: (i) Show that local connectedness cannot be replaced by connectedness. (ii) Show that if we replace the

R^2

plane with

R^3

space, the statement does not hold.

Problem Statement