= Radial extension of a square boundary homeomorphism
A <homeomorphism> $g:\partial Q\to\partial Q$ of the boundary of a closed square $Q$ extends to a <homeomorphism> of $Q$ by fixing its centre $c$ and setting $H(c+r(p-c))=c+r(g(p)-c)$ for $p\in\partial Q$ and $0<r\leq1$. Each noncentral point lies on a unique ray from $c$ meeting the boundary once, so the formula is well defined and its inverse is obtained from $g^{-1}$. Both maps are <continuous> off the centre by the radial parametrization, and at the centre because their distances from $c$ are bounded by $r$ times the radius of the square. It follows that complements of any two closed boundary arcs are <homeomorphic>: first map the arcs to one another by a boundary <homeomorphism>, then extend radially.
Back to article page
