How to make displaystyle unions, intersections, sums products, etc. less obnoxious

Question

When one tries to display, for example, a large union using ‘bigcup’ one issue is that text above and below gets pushed up and down, respectively. The below image shows that it is possible to correct this. (The ‘big cup’ is clearly pushed upwards, somehow, and it looks like it fits into the writing almost perfectly, without having any substantial affect on the line spacing.) My question is: how can I produce a similar result as the below, also for products and the like? Do I have to increase the size of the line spacing? Thank you.

\documentclass[class=book,crop=false,a4paper, 9pt, oneside, leqno]{standalone}
\usepackage[T1]{fontenc} 
\usepackage[utf8]{inputenc}        
\setlength{\baselineskip}{5in}
\usepackage[thmmarks, thref]{ntheorem}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{yhmath}
\begin{document}
An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example $\bigcup\limits_{s\in \mathscr{S}}\mathrm{W}_s(\mathrm{D})$ An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example An example. 
\end{document}

Answer 1

The commands \limits and \nolimits are placed immediately after variable-sized operator symbols, such as those produced by \int, \sum and \bigcup. \limits instructs TeX to place the subscript and superscript terms of the operation (the lower and upper limits of integration in the case of \int) immediately below and above the operator symbol itself. In contrast, \nolimits instructs TeX to place the subscript and superscript terms to the lower right and upper right of the symbol, which conserves vertical space while consuming more horizontal space.

In mathematical typesetting, a very widely followed typographical convention is to generate the small variant of variable-size operator symbols and to employ \nolimits if TeX is in inline math mode. This convention helps assure that the spacing between lines in a paragraph can be fairly even if \sum, \int, and \bigcup instructions occur in a paragraph. Conversely, if TeX is in display math mode -- as is the case immediately after, say, \begin{equation} -- it is conventional to generate the large variants of the variable-sized operators and to employ \limits by default (except in the case of \int, \iint, and friends).

The defaults associated with typographical conventions can be overridden. Naturally, you override the defaults at your own peril. For more information on how one might override the defaults see, e.g., the posting Show inline math as if it were display math (and vice versa).

The upper half of the following screenshot consists of a single paragraph. The paragraph features a large and unsightly gap below the line that contains $\bigcup\limits_{s\in \mathscr{S}}...$. Importantly, no such gap occurs below the lines that contain either $\bigcup_{s\in \mathscr{S}}...$ or $\bigcup\nolimits_{s\in \mathscr{S}}...$. "Pushing the bigcup operator upwards," as you put it, in no way suffices to prevent the gap below that line -- unless one moved up the \bigcup symbol so much as to force a gap above the line.

In short: If you want to avoid creating unsightly gaps in a paragraph that contains \bigcup, do not use \limits.

\documentclass[class=book,crop=false,a4paper, 9pt, oneside, leqno]{standalone}
\usepackage[T1]{fontenc} 
\usepackage[utf8]{inputenc}        
\setlength{\baselineskip}{5in}
\usepackage[thmmarks, thref]{ntheorem}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{yhmath}
\begin{document}
An example An example An example An example An example An example 
An example An example An example An example An example An example 
An example $\bigcup_{s\in \mathscr{S}}\mathrm{W}s(\mathrm{D})$ 
An example An example An example An example An example An example 
An example $\bigcup\limits{s\in \mathscr{S}}\mathrm{W}s(\mathrm{D})$ 
An example An example An example An example An example An example 
An example An example An example An example 
$\bigcup\nolimits{s\in \mathscr{S}}\mathrm{W}s(\mathrm{D})$ 
An example An example An example An example An example An example. 
[\bigcup{s\in \mathscr{S}}\mathrm{W}s(\mathrm{D})]
and
[\bigcup\limits{s\in \mathscr{S}}\mathrm{W}s(\mathrm{D})]
and
[\bigcup\nolimits{s\in \mathscr{S}}\mathrm{W}_s(\mathrm{D})]
\end{document}