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### Abstract

The almost zero $\omega$-enumeration degrees are introduced in [1].
In the talk we shall present some properties of the almost zero degrees and
their role in the study of the $\omega$-enumeration degrees below the
first jump of $\mathbf{0}_\omega$ with respect to the relation
"$\preceq$",
where
\[
\mathbf{a}\preceq \mathbf{b}\iff (\exists n)
(\mathbf{a}^{(n)}\leq_\omega
\mathbf{b}^{(n)}).
\]
1. I. N. Soskov and H. Ganchev, {\it The jump operator on the
$\omega$-enumeration degrees}, to appear in Annals of Pure and
Applied Logic.