Discussion
Let \(X = \{(x, y) \in \mathbf{R}^{2} : (x - n)^{2} + y^{2} = 1, n \in \mathbf{Z}\}\) be the subspace of the Euclidean space \(\mathbf{R}^{2}\). Then :
Answer: B
X is connected, but not compact.
X is the union of the unit circles centred at \((n, 0)\), \(n \in \mathbf{Z}\). Adjacent circles, centred 1 apart with radius 1, intersect, so the union of this chain of connected sets is connected.
X is not compact because it is unbounded: it contains the points \((n, 0)\) for every integer \(n\). By the Heine-Borel theorem a compact subset of \(\mathbf{R}^{2}\) must be bounded. So the options calling X compact, or not connected, are wrong.
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