From a290d583ca3c4dfc39115068f209d64449c93a03 Mon Sep 17 00:00:00 2001
From: bnewbold
Date: Sun, 24 Jan 2010 03:39:25 0500
Subject: math fixes

math/topology.page  73 ++++++++++++++++++++++++++
1 file changed, 35 insertions(+), 38 deletions()
(limited to 'math/topology.page')
diff git a/math/topology.page b/math/topology.page
index 6f03eee..ea369fb 100644
 a/math/topology.page
+++ b/math/topology.page
@@ 1,36 +1,33 @@
====================
Topology
====================
.. warning:: Incomplete; in progress

.. note:: Most of the definitions and notation in the section are based on [munkres]_
+*References: 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.
+concept of open and closed subsets on the real number line (such as $(0,1)$ and
+$[1,2]$) 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:
+Formally, a *topology* on a set $A$ is a collection $\mathcal{T}$ of
+subsets of $A$ fufiling the criteria:
 1. The empty set and the entire set :m:`$A$` are both in :m:`$\mathcal{T}$`.
+ 1. The empty set and the entire set $A$ are both in $\mathcal{T}$.
 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is
 also in :m:`$\mathcal{T}$`.
+ 2. The union of an arbitrary number of elements of $\mathcal{T}$ is
+ also in $\mathcal{T}$.
 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is
 also in :m:`$\mathcal{T}$`.
+ 3. The intersection of a finite number of elements of $\mathcal{T}$ is
+ also in $\mathcal{T}$.
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:`$\mathcal{T}$`.
+If a subset $B$ of $A$ is a member of $\mathcal{T}$ then
+$B$ is an open set under the topology $\mathcal{T}$.
*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:`$\mathcal{T'\in T}$`.
+$\mathcal{T'}$ is finer than $\mathcal{T}$ if $\mathcal{T}$
+is a subset of $\mathcal{T'}$ (and $\mathcal{T}$ is coarser);
+it is *strictly finer* if it is a proper subset (and $\mathcal{T}$ is
+*strictly coarser*). Two sets are *comprable* if either $\mathcal{T\in T'}$
+or $\mathcal{T'\in T}$.
*Smaller* and *larger* are somtimes used instead of finer and coarser.
Topologies can be generated from a *basis*.
@@ 42,40 +39,40 @@ Frequently Used Topologies
*Standard Topology*
The standard topology on the real line is generated by the collection of all intervals
 :m:`$$(a,b)=\{xa