We defined what a topology is, now we will discuss how to make "new topologies from old": the constructions for "subspace topologies", "quotient topologies" and next week we will look at "product topologies".

Most importantly,

• this second construction can be used to put a topology on a set of disjoint subsets/equivalence classes of a set, determined by a partition or an equivalence relation.
---> this allows for defining many "famous" top spaces, such as the Mobius strip, torus, Klein bottle, real projective plane.
• the same construction can be used to make the term "gluing" (or "pasting", "identifying") precise
---> for example, often, topologists talk of gluing points together. If X is a topological space and points x,y in X are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation a ~ b if and only if a = b or a = x, b = y (or a = y, b = x). The two points are henceforth interpreted as one point.
• As the reverse of "gluing/pasting" one obtains "cutting".
• Finally, cuttings and gluings/pasting allow for "cut-and-paste arguments", when one rearranges a space by using these two operations while keeping track of original identifications with the aim of being able to "see" the structure of the given space "better".
--->For example, a typical "cut and paste argument" shows that the real projective plane can be obtained from the Mobius strip by gluing a disk onto its bounding circle.

The summary for product topologies will be posted later