Explicit homeomorphism between the plane and the plane minus a closed half-line.
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Let $r$ be a closed half-line in $mathbb{R}^2$ and $delta$ a positive real number. I want to find an explicit homeomorphism $phi:mathbb{R}^2tomathbb{R}^2-r$ such that $phi(X)=X$ for all $X$ such that $d(X,r)geqdelta.$ I can suppose the half-line is ${(x,y)inmathbb{R}^2:xgeq0,y=0}.$ Moreover, if we put $$B={Xinmathbb{R}^2:d(X,r)<delta},$$ we must have $phi(B)=B-r.$ I don’t know how to write this homeomorphism. I thought I can deform this initial configuration into something other, but I don’t have any other ideas.
general-topology
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edited Dec 28 '18 at 4:56
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