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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.
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