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====================
Topology
====================

.. note:: Incomplete; in progress

.. note:: 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 :m:`$(0,1)$` and
:m:`$[1,2]$`) are generalized to arbitrary sets.

Formally, a *topology* on a set :m:`$A$` is a collection :m:`$\mathcal{T}$` of 
subsets of :m:`$A$` fufiling the criteria:

 1. The empty set and the entire set :m:`$A$`:m: are both in :m:`$\mathcal{T}$`.

 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is 
    also in :m:`$\mathcal{T}$`.

 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is
    also in :m:`$\mathcal{T}$`.

If a subset :m:`$B$` of :m:`$A$` is a member of :m:`$\mathcal{T}$` then
:m:`$B$` is an open set under the topology :m:`$\mathcal{T}$`.

*Coarseness* and *Fineness* are ways of comparing two topologies on the same space.
:m:`$\mathcal{T'}$` is finer than :m:`$\mathcal{T}$` if :m:`$\mathcal{T}$`
is a subset of :m:`$\mathcal{T'}$` (and :m:`$\mathcal{T}$` is coarser); 
it is *strictly finer* if it is a proper subset (and :m:`$\mathcal{T}$` is 
*strictly coarser*). Two sets are *comprable* if either :m:`$\mathcal{T\in T'}$`
or :m:`$\mathcal{T'\in T}$`.
*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 
    :m:`$$(a,b)=\{x|a<x<b\}$$`
    This is the usual definition for open sets on the real line.

*Discrete Topology*
    The topology on a set :m:`$A$` consisting of all points of :m:`$A$`;
    in other words the power set of :m:`$A$`.

*Trivial/Indiscrete Topology*
    The topology on a set :m:`$A$` consisting of only the empty set and :m:`$A$`
    itself. Not super interesting but it's always there (when :m:`$A$` isn't empty).
    
*Finite Complement Topology* (:m:`$\mathcal{T_f}$`)
    The topology on a set :m:`$A$` consisting of the empty set any subset 
    :m:`$U$` such that :m:`$A-U$` has a finite number of elements.

*Lower Limit Topology* (:m:`$\mathbb{R}_\mathcal{l}$`)
    The lower limit topology on the real line is generated by the collection of all half open
    intervals
    :m:`$$[a,b)=\{x|a\leq x<b\}$$`
    :m:`$\mathbb{R}_\mathcal{l}$` is strictly finer than the standard topology and
    is not comprable to :m:`$\mathbb{R}_K$`.

*K-Topology* (:m:`$\mathbb{R}_K$`)
    Let :m:`$K$` denote the set of all numbers :m:`$1/n$` where :m:`$n$` is
    a positive integer. 
    The K-topology on the real line is generated by the collection of all standard open intervals 
    minus :m:`$K$`. 
    :m:`$\mathbb{R}_K$` is strictly finer than the standard topology and is not
    comprable to :m:`$\mathbb{R}_\mathcal{l}$`.

*Order Topology*
    TODO



.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.