Discussion
Let X be a connected metric space. Then which of the following is true ?
Answer: C
The true statement is \(|X| \neq 10\).
In a metric space every finite set is closed, and every single point of a finite space is also open (its complement is a finite union of closed points). So a finite metric space with two or more points splits into open, closed singletons and is disconnected. A connected metric space is therefore either a single point or infinite; in fact a connected metric space with two points \(a \neq b\) contains a point at every distance \(r \lt d(a, b)\) from \(a\), so it is uncountable.
Either way it cannot have exactly 10 points. 'X is finite' fails for the real line, 'X is infinite' and 'X is uncountable' fail for a one-point space, so only \(|X| \neq 10\) holds for every connected metric space.
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