Is the circle homeomorphic to the parabola in $\mathbb{R}^2$? If yes/no then justify. I guess the circle is not homeomorphic to the parabola because the circle is compact in $\mathbb{R}^2$ but the parabola is not. Is this a sufficient justification?
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Yes, it is.${}{}$ – Daniel Robert-Nicoud Oct 07 '15 at 16:41
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3Your compactness argument is good. An alternative is to use connectedness: removing a point from the parabola disconnects it but removing a point from the circle does not. – Rob Arthan Oct 07 '15 at 16:45
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Thank you a lot for the answer. – Kavita Oct 07 '15 at 16:56