\chapter{Point-set topology}
The notion of a \emph{topology} is used widely in math,
but different areas use \emph{very} different aspects.
\section{Definition}
The notion of a topological space (a topology on a set) is an \emph{algebraic}
generalization of the notion of closed and open sets from metric topology.
It is a structure on a set.
We look at the algebraic properties that the set of closed sets
(dually, open sets) satisfies, and define a \Def{topology} on a set $X$
to be any set of subsets of $X$ that satisfy the same properties.
It is elegant but very confusing to write things in terms of the topology,
as the ``set of subsets'' above hints at.
(a continuous function $f\from (X,\tau) \to (X,\tau')$
is one such that $f^*\tau' \subset \tau$)
almost always you talk in terms of open and closed sets.
\section{Intuition from metric topology}
*closure*
$X \mapsto \bar X$
Why closure?
- constant sequence
- diagonalization
(people often start with open sets,
which are a bit less natural;
open sets are ones on which you can define derivatives)
empty/whole:
vacuously true, but also technically useful:
intersection of any 2 closed sets is closed
(what if they don't intersect? messy otherwise)
\subsection{Confusion: closed $\neq$ not open}
not like a door
beware of "half-open" intuition; look in R^2 instead.
\subsection{A metric is \emph{more} information than a topology}
also a uniform structure
topologies coming from metrics are special (separation, etc.)
\section{Algebraic intuition on lattices}
complement on $\cP(X)$:
antitone (order reversing),
top and bottom reverse,
intersection and union switch (De Moivre's theorems)
\section{Examples and applications}
Analysis? (different topologies on a given set! Weak^* topology)
[C^k is closed in the C^k metric/topology. unsurprisingly]
Algebraic Geometry? (Sheaves and Spec?, etale topology)
Manifold Topology? (not much: all your spaces are nice)
all you care about are "connected, compact, interior, boundary"
ok, you also care about functional analysis
Algebraic Topology? (categoricalness: really just need a "good category")
Logic / Topos theory? (locales, models of constructive logic)
pointless topology:
don't need points!
just need lattice!!
\section{New spaces for old}
Given sub*sets*, you get sub*spaces* by restricting sets
subset
(Given $A \to X$, topology on $A$ is $f^*\tau$)
yes, this is an adjunction, I think;
hell, it's universal
quotient: very subtle!
product
(infinite products have issues)
coproduct
gluing;
formally, pushout (!)
[also "coequalizer" if gluing space to itself]
fiber bundles
pullbacks