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.