Discussion
Consider the following statements :
(a) Every non-empty open subset of R is uncountable.
(b) Every connected subset of R containing at least two elements is uncountable.
Then :
Answer: C
Both (a) and (b) are true.
(a) A non-empty open subset of R contains an open interval \((x - \varepsilon, x + \varepsilon)\), and every non-degenerate interval is uncountable, so the set is uncountable.
(b) The connected subsets of R are exactly the intervals. A connected set containing two points \(a \lt b\) therefore contains the whole interval \([a, b]\), which is uncountable.
Since both statements hold, the options saying only one is true, or both false, are wrong.
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