In Trefethen Bau Numerical Algebra, floating point set F is defined by
$\textbf{F} = \left\{\pm(m/\beta^{t})\beta ^{e}| 1 \leq m \leq \beta^{t}, e \in \mathbb{Z} \right\}$
Equivalently, by making $m$ 1 to 1 correspond to each floating points,
$\textbf{F} = \left\{\pm(m/\beta^{t})\beta ^{e}| \beta^{t-1} \leq m \leq \beta^{t} - 1, e \in \mathbb{Z} \right\}$
So, the gap size between each floating point I think is $\beta ^{e-t}$, which makes machine epsilon be $\frac{1} 2 \beta ^{-t}$ (because $1$ in F by the first definition is when $m = \beta ^{t}$ and $e = 0$). And the second definition is telling that it is $\frac{1} 2 \beta ^{1-t}$, which corresponds to the definition of machine epsilon in the book. But the gap size is still unchanged.
1. Other resources on the Internet told that the gap size of floating point is $\beta ^{1-t+e}$. What am I wrong?
2. Why are the machine epsilons obtained by each definition different?
In first definition, there are many 1's by different m and e. Which one should I choose? Should I just use 2nd def?