We say $A$ is a $2$-distance set if there are numbers $a,b$ such that the distance between two distinct points of $A$ can only take the value $a$ or $b$.
Are there any uncountable $2$-distance sets in $l_2$?
Can there be a $2$-distance set of uncountable cardinality in $l_2$?
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3By $l_2$, you refer to the space of square-summable sequences, presumably, and not to $\ell^2(S)$ for some other set. Then $l_2$ is second countable, hence every uncountable subset has (uncountably many) limit points. – 2017-01-14
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A $2$-distance set is always discrete and closed. If $a$ is the smaller of the two possible distances, every open ball of radius $\leqslant a/2$ contains at most one point of the $2$-distance set.
Hence in every second countable (equivalently, separable) metric space, a $2$-distance set can be at most countable.
Since the space of square-summable sequences is second countable, it cannot contain an uncountable $2$-distance set. But for every uncountable set $S$, the Hilbert space $\ell^2(S)$ contains uncountable $2$-distance sets. Also, the sequence space $\ell^{\infty}(\mathbb{N})$ contains uncountable $2$-distance sets.