Hello Minhyong. I think other people have already given several excellent answers about how to motivate topology, and I'm not sure I have much to add. But there are a couple of other parts to your question. I had the impression that the notion of a topological space was introduced by Hausdorff in his 1914 book on set theory (Mengenlehre). But my history is somewhat shaky, and it would be nice if someone else could confirm this. It certainly seems that by the early 20th century there was a radical shift in the way mathematics was done.

Jumping ahead to your added comment, I agree that a Grothendieck topology is a very natural extension of this idea. However, for Grothendieck this was secondary; the associated category of sheaves or topos was the important thing. Of course, you know that, but perhaps not everyone does.