Formal definitions of cutting and pasting etc

[navigator]

Popular math books usually explain topology as the study of spaces allowing strecthing and shrinking and some cutting and pasting, but what the formal definitions of the intuitive operations "cutting","pasting","strecthing","shrinking" according to mathematics?

 

[quasar987]

cutting is a set operation. S\A means "The set S with the set A removed (or "cut") from it."

pasting is the operation of taking the connected sum. You can read about it on wikipedia.

stretching and shrinking are the result of applying a certain map called a homeomorphism to your surface.

 

[HallsofIvy]

Very roughly speaking, stretching and shrinking involve continuous changes while cutting and pasting involve non-continuous changes. Fundamentally, "topology" is the study of continuity.

 

[mrandersdk]

The thing you are looking for is algebraic topology, there is a free book here

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

even though it seems very simple it is hard to describe it formaly.

 

[navigator]

But when you cut a curve off, you just partition the curve into two segments which is not connected and no point (even the cut point) is been removed, this is somewhat quite difference from complement operation that remove the elements from the universal set, so i think the best way to define it is $$A \times {0} \cup B \times {1}$$, do you agree? but the problem appearring, is if we only mention the curve and the cut point, how could we do?

 

[quasar987]

I didn't know you were talking about "separating" a curve like that. I though you were talking about the operation of cutting a little piece off a surface before gluing-in another surface.

More accurately I should have said that the operation of "cutting & pasting" considered as a whole is called taking the connected sum of two surfaces (manifolds). Cutting itself consists of removing a small ball from each surface (which does corresponds to the set operation S\A) and pasting is the introduction of an equivalence relation on the two surfaces that declare equivalent the points on the boundaries where the balls were removed.