Discussion
Consider the set :
\[ A = \{(x, y) \,/\, xy = 0\} \]
Which of the following is false ?
Answer: D
The false statement is A is not connected.
The set \(xy = 0\) is the union of the two coordinate axes. It is the zero set of the continuous function \(xy\), so it is closed in \(\mathbf{R}^{2}\).
Any point on the x-axis can be joined to the origin along that axis, and the origin to any point on the y-axis along the other, so A is path connected and hence connected. That makes 'A is not connected' the false statement.
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