Topologists often mention an example beginning by "If there is no tear and no paste, then ...". As a student, I am confused with this "term", and I want to know the exact mean of it.
First of all, what are tear and paste? Assume there is a quotient map from $X$ to $Y$, then can we call $Y$ is pasted by $X$, and $X$ is teared from $Y$?
I have searched for http://en.wikipedia.org/wiki/Ambient_isotopy. I think the term "no tear and no paste" may be related with an ambient space, but I cannot figure out what its exact mean is or what the relationship is between the term and the concept of paste and tear.
Any advice is helpful. Thank you.
"No tearing and no pasting" are not technical terms and do not have any precise meaning. You will almost never find them in any text written by mathematicians for mathematicians. They are instead informal terms which are sometimes used to convey an intuitive idea to laypeople without getting into any technical definitions. Typically, they correspond to something involving continuity, but the exact technical meaning that the phrase is concealing varies with context.