In the book Topology and Geometry by Bredon, the following is offered as an example of a topological space on page 6 in chapter 1:
$X = \omega \cup \{\omega\}$ with the open sets being all subsets of $\omega$ together with complements of finite sets. (Here, $\omega$ denotes the set of natural numbers.)
I am unsure of the reason for including the nested set $\{\omega\}$ within this definition. Is this standard notation? Does the author truly mean that $X=\{\omega,1,2,3,...\}$? If so, why is the inclusion of $\omega$ important for the construction?