Discussion
The interval (−1, 1) is homeomorphic to :
Answer: A
(−1, 1) is homeomorphic to the parabola \(y = x^{2}\).
The parabola is the graph of a continuous function on R, so \(x \mapsto (x, x^{2})\) is a homeomorphism from R onto it. And (−1, 1) is homeomorphic to R, for example by \(x \mapsto \tan\frac{\pi x}{2}\).
The others are distinguished by topological invariants. The ellipse and the circle are compact, while (−1, 1) is not. [−1, 1] is compact too. Removing a point from the circle or ellipse leaves it connected, but removing a point from an open interval disconnects it.
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