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Investigate whether the following set is closed or open in $\mathbf{R}^2$. Provide precise justification.

$$A = \{(x, y) \in \mathbf{R}^2 : y^2 = x\}.$$

To show that $A^C$ is open, which is equivalent to demonstrating that $A$ is closed, I tried to utilize the theorem stating that $A^C$ is open if and only if there exists a sequence $\{a_n\}_{n \in \mathbb{N}} \subset \mathbf{R}^2$ converging to $L$, where $L$ is not in $A^C$. Thus, $A^C$ is closed.

Could someone provide an approach to define the sequence, or suggest an easier method to tackle this problem? Any assistance would be greatly appreciated.

  • Pick a point not on the parabola and show there is an open disk centered at this point that does not meet the parabola. – Chris Leary Mar 18 '24 at 20:58
  • Your formulation of the sequence theorem seems wrong. However, you can still utilize sequences here. $A$ is closed if for every convergent sequence ${a_n}_{n \in \mathbb{N}}$ in $A$, it has its limit in $A$. Use this along with the fact that the squaring function is continuous – C Squared Mar 18 '24 at 21:02
  • None of the linked duplicates are exactly the same question, but the techniques described in the answers are applicable here, and answer the question. – Xander Henderson Mar 18 '24 at 21:12

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