From 0b113afaa8ae15d3672c6c50f6f6b5b26d78d618 Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@eta.mit.edu>
Date: Wed, 5 Nov 2008 02:26:39 -0500
Subject: better for now

---
 math/topology | 54 +++++++++++++++++++++++++++---------------------------
 1 file changed, 27 insertions(+), 27 deletions(-)

(limited to 'math')

diff --git a/math/topology b/math/topology
index 6b44484..79beeae 100644
--- a/math/topology
+++ b/math/topology
@@ -17,20 +17,20 @@ 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 `$\mathcal{T}$`:m:.
+    also in :m:`$\mathcal{T}$`.
 
- 3. The intersection of a finite number of elements of `$\mathcal{T}$`:m: is
-    also in `$\mathcal{T}$`:m:.
+ 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is
+    also in :m:`$\mathcal{T}$`.
 
-If a subset `$B$`:m: of `$A$`:m: is a member of `$\mathcal{T}$`:m: then
-`$B$`:m: is an open set under the topology `$\mathcal{T}$`:m:.
+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.
-`$\mathcal{T'}$`:m: is finer than `$\mathcal{T}$`:m: if `$\mathcal{T}$`:m:
-is a subset of `$\mathcal{T'}$`:m: (and `$\mathcal{T}$`:m: is coarser); 
-it is *strictly finer* if it is a proper subset (and `$\mathcal{T}$`:m: is 
-*strictly coarser*). Two sets are *comprable* if either `$\mathcal{T\in T'}$`:m:
-or `$\mathcal{T'\in T}$`:m:.
+: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*. 
@@ -41,35 +41,35 @@ Frequently Used Topologies
 
 *Standard Topology*
     The standard topology on the real line is generated by the collection of all intervals 
-    `$$(a,b)=\{x|a<x<b\}$$`:m:
+    :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 `$A$`:m: consisting of all points of `$A$`:m:;
-    in other words the power set of `$A$`:m:.
+    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 `$A$`:m: consisting of only the empty set and `$A$`:m:
-    itself. Not super interesting but it's always there (when `$A$`:m: isn't empty).
+    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* (`$\mathcal{T_f}$`:m:)
-    The topology on a set `$A$`:m: consisting of the empty set any subset 
-    `$U$`:m: such that `$A-U$`:m: has a finite number of elements.
+*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* (`$\mathbb{R}_\mathcal{l}$`:m:)
+*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
-    `$$[a,b)=\{x|a\leq x<b\}$$`:m:
-    `$\mathbb{R}_\mathcal{l}$`:m: is strictly finer than the standard topology and
-    is not comprable to `$\mathbb{R}_K$`:m:.
+    :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* (`$\mathbb{R}_K$`:m:)
-    Let `$K$`:m: denote the set of all numbers `$1/n$`:n: where `$n$`:m: is
+*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 `$K$`:m:. 
-    `$\mathbb{R}_K$`:m: is strictly finer than the standard topology and is not
-    comprable to `$\mathbb{R}_\mathcal{l}$`:m:.
+    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
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