From a290d583ca3c4dfc39115068f209d64449c93a03 Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@ziggy.(none)>
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:`$$(a,b)=\{x|a<x<b\}$$`
+    $$(a,b)=\{x|a<x<b\}$$
 
     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:`$A$`.
+    The topology on a set $A$ consisting of all points of $A$;
+    in other words the power set of $A$.
 
 *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).
+    The topology on a set $A$ consisting of only the empty set and $A$
+    itself. Not super interesting but it's always there (when $A$ 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.
+*Finite Complement Topology* ($\mathcal{T}_f$)
+    The topology on a set $A$ consisting of the empty set any subset 
+    $U$ such that $A-U$ has a finite number of elements.
 
-*Lower Limit Topology* (:m:`$\mathbb{R}_{\mathcal{l}}$`)
+*Lower Limit Topology* ($\mathbb{R}_{\mathcal{l}}$)
     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:`$\mathbb{R}_K$`.
+    $$[a,b)=\{x|a\leq x<b\}$$
+    $\mathbb{R}_{\mathcal{l}}$ is strictly finer than the standard topology and
+    is not comprable to $\mathbb{R}_K$.
 
-*K-Topology* (:m:`$\mathbb{R}_K$`)
-    Let :m:`$K$` denote the set of all numbers :m:`$1/n$` where :m:`$n$` is
+*K-Topology* ($\mathbb{R}_K$)
+    Let $K$ denote the set of all numbers $1/n$ where $n$ 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:`$\mathbb{R}_{\mathcal{l}}$`.
+    minus $K$. 
+    $\mathbb{R}_K$ is strictly finer than the standard topology and is not
+    comprable to $\mathbb{R}_{\mathcal{l}}$.
 
 *Order Topology*
     TODO
 
 
 
-.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.
+[^munkres] **Topology (2nd edition)**, by James R. Munkres.
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