Discussion
With the induced topology by the metric \(d(x, y) = |x - y|\),
Answer: B
\(\mathbf{R}\) is of second category and \(\mathbf{Q}\) is not.
\(\mathbf{R}\) with the usual metric is complete, so by the Baire category theorem it is not a countable union of nowhere dense sets: it is of second category in itself.
\(\mathbf{Q}\) is countable, and each singleton \(\{q\}\) is nowhere dense in \(\mathbf{Q}\) (a single point has empty interior there). So \(\mathbf{Q} = \bigcup \{q\}\) is a countable union of nowhere dense sets, of first category. That rules out the options calling \(\mathbf{Q}\) second category or denying it to \(\mathbf{R}\).
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