I am reading a paper in which a curve is referred to as being '$\epsilon$-close to a horizontal line in the $C^{1/\epsilon}$-topology'. There is no explanation of what this means, and as an undergraduate student I have not come across this topology (or even know if it's standard notation). The concept of a Whitney topology has similar notation but seems only defined for integers and I'm not sure whether this notation is just a coincidence. Has anyone come across this and could please explain it? A google search has returned nothing. Thanks!
Edit: didn't include the paper as the above is mentioned as a comment rather than part of the main text. But here it is for those interested, remark 4.2 on page 9.