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 
    $$(a,b)=\{x|a<x<b\}$$

    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 $A-U$ 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
    $$[a,b)=\{x|a\leq x<b\}$$
    $\mathbb{R}_{\mathcal{l}}$ is strictly finer than the standard topology and
    is not comprable to $\mathbb{R}_K$.

*K-Topology* ($\mathbb{R}_K$)
    Let $K$ denote the set of all numbers $1/n$ where $n$ is
    a positive integer. 
    The K-topology 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.