Universal uniqueness criterion?

[Kouznetsov]

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And that implies that \( f(S) \), which contains an open non-empty set, always contains points \( w \) such that \( \lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty \).
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Why do you think that this set is open? Wiki says:
In metric topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U. In other words, the distance between any point x in U and the edge of U is always greater than zero.
http://en.wikipedia.org/wiki/Open_set

The small deviation from a point where the limit is finite allows to get (in the limit) so high walues as you want, because the right hand side of the plot of tetration is pretty scratched. I would replace "open non-empty set" to "non-empty set".