Discussion
Consider the topology on \(\mathbb{Z}\) with basis \(\{S(a, b) : a, b \in \mathbb{Z}\) and \(a \neq 0\}\), where
\(S(a, b) = \{an + b : n \in \mathbb{Z}\}\).
Consider the following statements.
I. \(S(a, b)\) is both open and closed for each \(a, b \in \mathbb{Z}\) with \(a \neq 0\).
II. The only connected set containing \(x \in \mathbb{Z}\) is \(\{x\}\).
Which one of the following is correct?
Answer: A
Both I and II are TRUE.
This is Furstenberg's topology on \(\mathbb{Z}\). I: \(S(a, b)\) is open as a basis set, and its complement is the union of the other residue classes \(S(a, b + 1), \ldots, S(a, b + |a| - 1)\), which are open; so \(S(a, b)\) is also closed.
II: take any set \(C\) containing \(x\) and another point \(y \ne x\). Choose \(a\) with \(a \nmid (y - x)\); then \(S(a, x)\) is clopen, contains \(x\) and not \(y\), and splits \(C\) into two disjoint relatively open pieces. So no connected set has more than one point. Hence options with I or II false are wrong.
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