Discussion
Let \(A = \{(x, y) \in \mathbf{R}^{2} : x, y \text{ are integers}\}\) and \(B = \{(x, y) \in \mathbf{R}^{2} : y = x^{2}\}\). Then :
Answer: B
\(\mathbf{R}^{2} - A\) is connected.
A is the integer lattice, a countable closed discrete set. Removing it from the plane leaves a path connected set: any two points can be joined by a polygonal path that dodges the lattice points, for instance by moving along horizontal and vertical lines at non-integer heights and abscissas.
The parabola B behaves differently. \(\mathbf{R}^{2} - B\) splits into the two regions \(y \gt x^{2}\) and \(y \lt x^{2}\), and any path between them must cross the parabola by the intermediate value theorem applied to \(y - x^{2}\). So it is not connected.
Neither complement is closed: A and B are closed and nonempty, so their complements are open proper subsets of the connected plane, and such sets are never closed.
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