Why we need topology to define these new form open sets? By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms of open sets.) But I think in the practical cases, we just considered Euclidean space most and the traditional form open set in Euclidean space works pretty well. And these new form open sets seem to have no common use. So why we need to expand the conception of the open sets? What's the motivation behind?Thanks.
I think your idea that topology is just to define open sets misses the point. I interpret the point of topology as attempting to generalize the concept of a continuous function primarily, and open sets are often the most direct pedagogical route there. 

Further, one often does want to leave euclidean space. For example, we often want to consider spaces of continuous functions where the topology (say, pointwise convergence) may not even be metrizable! Topological spaces that are not first countable, and thus not metrizable, also arise quite naturally in functional analysis.

Even at times when one can fundamentally imbedd the space being worked on into some euclidean space, this is often a cumbersome process adding on unnecessary baggage.
We need topology so we can work limits. And limits are important.