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".
Here is a
summary
of "subspace topologies" and the related "inclusion function"
.
Here is
a
summary for quotient topologies and
the related quotient
function.
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