From f61026119df4700f69eb73e95620bc5928ca0fcb Mon Sep 17 00:00:00 2001
From: User
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