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Topology
====================
*References: Most of the definitions and notation in the section are based on [^munkres]*
A *topological space* is a set for which a valid topology has been defined: the topology
determines which subsets of the topological space are open and closed. In this way the
concept of open and closed subsets on the real number line (such as $(0,1)$ and
$[1,2]$) are generalized to arbitrary sets.
Formally, a *topology* on a set $A$ is a collection $\mathcal{T}$ of
subsets of $A$ fufiling the criteria:
1. The empty set and the entire set $A$ are both in $\mathcal{T}$.
2. The union of an arbitrary number of elements of $\mathcal{T}$ is
also in $\mathcal{T}$.
3. The intersection of a finite number of elements of $\mathcal{T}$ is
also in $\mathcal{T}$.
If a subset $B$ of $A$ is a member of $\mathcal{T}$ then
$B$ is an open set under the topology $\mathcal{T}$.
*Coarseness* and *Fineness* are ways of comparing two topologies on the same space.
$\mathcal{T'}$ is finer than $\mathcal{T}$ if $\mathcal{T}$
is a subset of $\mathcal{T'}$ (and $\mathcal{T}$ is coarser);
it is *strictly finer* if it is a proper subset (and $\mathcal{T}$ is
*strictly coarser*). Two sets are *comprable* if either $\mathcal{T\in T'}$
or $\mathcal{T'\in T}$.
*Smaller* and *larger* are somtimes used instead of finer and coarser.
Topologies can be generated from a *basis*.
TODO: Hausdorf
Frequently Used Topologies
============================
*Standard Topology*
The standard topology on the real line is generated by the collection of all intervals
$$(a,b)=\{xa<x<b\}$$
This is the usual definition for open sets on the real line.
*Discrete Topology*
The topology on a set $A$ consisting of all points of $A$;
in other words the power set of $A$.
*Trivial/Indiscrete Topology*
The topology on a set $A$ consisting of only the empty set and $A$
itself. Not super interesting but it's always there (when $A$ isn't empty).
*Finite Complement Topology* ($\mathcal{T}_f$)
The topology on a set $A$ consisting of the empty set any subset
$U$ such that $AU$ has a finite number of elements.
*Lower Limit Topology* ($\mathbb{R}_{\mathcal{l}}$)
The lower limit topology on the real line is generated by the collection of all half open
intervals
$$[a,b)=\{xa\leq x<b\}$$
$\mathbb{R}_{\mathcal{l}}$ is strictly finer than the standard topology and
is not comprable to $\mathbb{R}_K$.
*KTopology* ($\mathbb{R}_K$)
Let $K$ denote the set of all numbers $1/n$ where $n$ is
a positive integer.
The Ktopology on the real line is generated by the collection of all standard open intervals
minus $K$.
$\mathbb{R}_K$ is strictly finer than the standard topology and is not
comprable to $\mathbb{R}_{\mathcal{l}}$.
*Order Topology*
TODO
[^munkres] **Topology (2nd edition)**, by James R. Munkres.
