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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 `$\mathcal{T}$`:m:.

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

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

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

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

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

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

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

*Order Topology*
    TODO



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