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.
For starters, we have things defined abstractly (at least in principle) such as manifolds. These are defined through charts, which endow your manifold with a topology.
The important tool of partitions of unity (which reduces Stokes's theorem and others to a toy case) works due to the 2nd Countability Axiom (see e.g. Warner, Foundations of Differential Manifolds and Lie groups).
There are important results in Commutative Algebra such as finiteness of irreducible components of the spectrum of a noetherian commutative ring: topology furnishes a proof of an algebraic result (see Atiyah & Macdonald, Introduction to Commutative Algebra).
Also, the topology is kind of qualitative, i.e. you are not considering a particular metric only, but only those equivalent to it (i.e. those which yield the same topology).
There are many more things one could say, but we'll leave it at that until further comments appear.
Where you can see a nice interplay between topology and analysis is in the Gelfand-Naimark theorem: the maximal rings of the ring of (real or complex) continuous functions over a compact topological space $K$ is homeomorphic to $K$ itself.
Again, a good reference is Atiyah-Macdonald (it's in the Exercises section, if I remember rightly).
We need topology so we can work limits. And limits are important.