 Solution

The subspace opens are $A\cap U$; quotient opens are those whose inverse image is open; a homeomorphism is a continuous bijection with continuous inverse. Here $q(x,y)=(|x|,y)$ identifies exactly the required pairs and induces a homeomorphism $Y\to [0,1]\times [-1,1]$. An affine rescaling maps that rectangle homeomorphically onto $S$.