Unknown notation/omega

Question

What does the 3rd term in $\omega\times\omega\times\omega^{\operatorname*{\omega}\limits_{\smile}}$ with semicircle below the last $\omega$ in definitions 5.3.6 here mean?

Answer 1

On p.4 ("0. Introduction", subsection "0.2 Notation"), you have in item 6.

The collection of all finite partial functions from $\omega$ to $X$ is $X^{\operatorname*{\omega}\limits_{\smile}}$.