
What Is Topology? - laronian
https://medium.com/cantors-paradise/what-is-topology-963ef4cc6365
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dwohnitmok
I've always found how topology and algebra treat the differences between
integers and rational numbers to be instructive at developing an intuition for
each.

As sets the integers and rational numbers are identical. They can be put in
one-to-one correspondence with each other.

To distinguish them further we must impose additional structure and examine
their differences.

Algebra is the study of operators over sets. This means from an algebraic
perspective the difference between the rational numbers and the integers are
the operators supported. The integers support addition, subtraction, and
multiplication. The rationals are formed by adding division.

On the other hand, topology is the study of how to formalize the intuitive
notions of "close" and "far." From a topological perspective, the
distinguishing feature of the integers is that each integer is very far from
the other, or equivalently that the integers are very sparse. Every integer
has two other integers between which there are no other integers. On the other
hand the rational numbers are very dense. They are the result of taking the
integers and "squishing" them together. Between and two rational numbers there
are an infinite number of other rational numbers.

~~~
SamReidHughes
The act of “squishing” wouldn’t affect order, so each squished integer would
still have two closest neighbors.

The set of rationals, Q, is homeomorphic to their Cartesian product, Q x Q. So
any analogy needs to work for that set, too.

~~~
dwohnitmok
That's fair. "Squishing the integers" was a poor choice of words. "Squishing"
an undistinguished countable set is perhaps a better analogy.

In particular I was thinking of undistinguished countable sets (although I was
confusingly throwing in ordering of the integers to try to make my point more
accessible) that you then add topological structure on top of.

In that world the integers are simply the discrete topology on a countable
set. Or more explicitly (to contrast with the next definition), where all
singleton sets are open.

The rationals then are formed by any metric whose induced topology does not
include singleton sets.

That is, any attempt to uniformly bring elements "closer" than the world where
single points are open gives rise to a topology homeomorphic to the rational
numbers.

~~~
SamReidHughes
The thing that surprised me yesterday (since I don't know much about topology)
is that every metrizable countable set without isolated points is the same.

So the numbers k/2^n, the points on the unit circle with rational y/x, or the
set Q \ Z, are homeomorphic.

The fact that you can slice and dice the rationals is counterintuitive.

~~~
SamReidHughes
Update: it's not counterintuitive any more.

