Let $X$ be the union of axes given by $xy = 0$ in $\Bbb R^2$ . Is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$?
If we remove the origin from the union of axes given by $xy = 0$ in $\Bbb R^2$, then we get $4$ connected pieces. Let if possible let there be an homeomorphism. But if we remove the image of the origin from the image sets, we get the following:
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One piece in case of the circle.
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Two pieces in case of a line.
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Two pieces in case of a parabola.
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Three pieces in case of rectangular hyperbola $xy = 1$.
Thus if true homeomorphism exists and since connectedness is a topological property then it would send four connected pieces of the union of axes given by $xy = 0$ in $\Bbb R^2$ to four connected pieces each of a line, a circle, a parabola or the rectangular hyperbola $xy = 1$, which is not true. So no homeomorphism exists.
Is the proof correct?
Best Answer
Yes, this is correct: $X$ is not homeomorphic to any of the choices, and your reasoning is correct. You can also rule out the rectangular hyperbola by noting that it is not connected, while $X$ is connected, and you can also rule out the circle by noting that it’s compact, while $X$ is not.