Why the word 'space' is used with metric space? I am new to math. According to the Wikipedia, A metric space is a set for which distances between all members of the set are defined. I have a silly question: Why they used the word 'space'? Why not 'metric set' for example? 
Many thanks for your answer in advance.

Generally speaking, good terminology should appeal to our intuition. Your idea of a "metric set" has that disadvantage that the word "set" applies to just about everything in mathematics and so it gives us no intuitive hook whatsoever. 
In the case of the word "space", this is a clue that there are geometric concepts lying around and that you should be prepared to engage your geometric intuition. 
So, for example, a metric space is founded on a formalization of the geometric concept of distance, and you should be prepared to engage your intuition about distance when reasoning about metric spaces. 
Other examples in the very nice answer of @ThomasAndrews contain different appeals to our geometric intuition: in a vector space we can employ our intuition about lines, planes, parallelograms etc.; in a measure space we can engage our intuition about area.
