For each space below, identify a familiar space homeomorphic to the quotient and prove your answer.
Let \(X=[0,1]\times[0,1]\), and let \(\sim\) be the smallest equivalence relation such that \[ (x,0)\sim(x,1),\qquad (0,y)\sim(1,y) \] for all \(x,y\in[0,1]\).
Let \(X=D_1\sqcup D_2\) be the disjoint union of two closed disks. Identify boundary points with the same Euclidean coordinates: for \((x,y)\in D_1\) and \((r,s)\in D_2\) with \(x^2+y^2=1\), impose \[ (x,y)\sim(r,s)\quad\Longleftrightarrow\quad(x,y)=(r,s). \]
Let \(X=D^2\) and collapse the entire boundary circle to one point; equivalently, let \(\sim\) be the smallest equivalence relation satisfying \[ (x,y)\sim(1,0) \] whenever \(x^2+y^2=1\).