Doubt on the definition of ordered topology given in 'Foundations of Topology By C. Wayne Patty'

Question

If the underlined symbol is as it is, The definition is confusing. Is it $\mathscr T$ or $\mathscr S$? Please help me with the definition.

Answer 1

All marked $\mathcal{T}$ should be $\mathcal{S}$, of course.

<rant> It's too bad so many math-books are submitted as pdf's from Latex sources without the need for human copy editors any more. Huge savings, I know. But earlier all published maths texts went through human eyes during proof reading and type setting and such errors were more rare. </rant>

Nitpick: note the proodf only works if there are at least two points in $X$. If there is one, so $X = \{p\}$ then $\mathcal{S} = \{\emptyset\}$ and that's a fine subbase (for the discrete = indiscrete topology on $X$) but not in this author's view.