Is this a Cantor Set? Nomenclature question.

Question

I can form the archetypal cantor set by removing the middle third of an interval, and recursively applying the same treatment to the two intervals remaining.

If instead I start with the interval $[0,1]$, and remove the middle fraction $f$ ($0<f<1$), then remove the middle fraction $f^2$ from the remaining two, the middle fraction $f^3$ from each of the remaining 4 and so on. I end up with a set of line segments of total measure $\exp[\sum_{n=1}^\infty \log(1-f^n)]$ (which converges). This seems to have some of the properties of the Cantor Set - it is a perfect set that is nowhere dense - but it is not measure zero. Is this a Cantor Set? Does this have a name?

Answer 1

When $f < \frac 13$ the resulting set is commonly called a "fat Cantor set". In general I have seen "Smith–Volterra–Cantor set" to refer to the resulting set, see e.g. this Wikipedia article.

Answer 2

At least in my field (dynamical systems) this would be called a Cantor set, yes. In fact we tend to still call "Cantor set" any topological space homeomorphic to the standard middle-third Cantor set.