Am I correct in stating that the study of topology is purely theoretical? Am I correct in stating that the study of topology is purely theoretical?
To clarify, the real world is discrete or quantized (ie; digital) whether we are discussing atoms, quarks, or strings, etc; but topology seems to depend upon everything being continuous or analog. 
For example, if I draw a one inch line and a two inch line on a  chalkboard, there is a topological mapping of every point on the one inch line to the two inch line, but in actual fact there are twice as many particles of chalk (or atoms, etc.) on the two inch line as there are on the one inch line.  What real world value then does the study of topology serve? 
Recommendations for books or publications that answer this fundamental question are welcome. 
 A: You have asked a philosophical question, not a mathematical one.
If by "purely theoretical" you mean something like "useless in the real world" then the answer is no. Topology has contributed significantly to our understanding of differential equations, which model many useful real world phenomena. In quantum mechanics, resolving the puzzling "wave particle duality" depends on seeing how the appropriate mathematical model can be treated with the mathematics of continuous systems. Without that kind of analysis we couldn't understand transistors and build computers. 
String theory (which has not yet been useful in the real world) relies on lots of topological ideas - in particular, on the study of Calabi-Yau manifolds. 
When you say 

the real world is discrete or quantized

you are on shaky philosophical ground. We live in the real world, but we don't know what it "is". All we have in physics are mathematical models we use to make predictions about how it behaves. At the present time the best mathematical models for its small scale behavior are quantum mechanical, but they don't say the real world is "really made up of discrete pieces". 
A: I would invite you to read Robert Ghrist's Elementary Applied Topology or check out Adams & Franzosa's Introduction to Topology: Pure and Applied to get a taste of some of the numerous applications of topology in the real world, regardless of whether the mathematical models involved match reality perfectly at all scales.
A: According to a report from the National Institutes of Health,

topological properties of the genetic material [...] influence virtually every major nucleic acid process.

One might be tempted to say that all life on earth depends on topology.
For more information, do a web search for "DNA topology".
A: Topology is possible to apply, but to apply it you need to know it, which not particularly many engineers do. :)
Same with any other higher math. You can do really cool applied abstract algebra things too, given you get guy responsible on project to thumb you up. This rarely seems to happen if they are not confident they understand it.
