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

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.
