From f61026119df4700f69eb73e95620bc5928ca0fcb Mon Sep 17 00:00:00 2001
From: User <bnewbold@daemon.robocracy.org>
Date: Tue, 13 Oct 2009 02:52:09 +0000
Subject: Grand rename for gitit transfer

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+====================
+Topology
+====================
+
+.. warning:: 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:`$A$` 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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