Thanks for all your guys reply. Actually, I am now more confused after the last bit of the discussion..... But, indeed, this is a question without a conclusive answer. So, I'd like to ask the following: sometimes I found the proof in topic of metric space is redundant and unnecessary; the result itself is intuitive and "obvious"( especially those about A map to B and bahbahbah property remains the same). In this case, is this a "good" habit for physicist to ignore the "minor" details of the vigorous proof or a harmful habit which will cause me losing some important knowledge for my future career? Thanks for everyone's contribution to this post !