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

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

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

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

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

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

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

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

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

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

*K-Topology* (`$\mathbb{R}_K$`:latex:)
    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:. 
    `$\mathbb{R}_K$`:latex: is strictly finer than the standard topology and is not
    comprable to `$\mathbb{R}_\mathcal{l}$`:latex:.

*Order Topology*
    TODO



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