On ℤ give the topology with basis S(a,b)={an+b:n∈ℤ}, a≠0. I: Every S(a,b) is both open and closed. II: The only connected set containing x is {x}. Choose truth values.
S(a,b) is a residue class modulo |a|. Its complement is the finite union of other residue classes, so it is clopen. For distinct x,y choose a modulus greater than |x−y|; the clopen class containing x excludes y and separates any subset containing both. Hence connected subsets are singletons. **A**.
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