From a2d7b8c246db71e6c06e9f3db267b5a98691228d Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@eta.mit.edu>
Date: Wed, 5 Nov 2008 02:18:57 -0500
Subject: fixed latex math?

---
 math/topology | 62 +++++++++++++++++++++++++++++------------------------------
 1 file changed, 31 insertions(+), 31 deletions(-)

(limited to 'math')

diff --git a/math/topology b/math/topology
index 104cbe8..c7e482f 100644
--- a/math/topology
+++ b/math/topology
@@ -8,29 +8,29 @@ Topology
 
 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 `$(0,1)$`:latex: and
-`$[1,2]$`:latex:) are generalized to arbitrary sets.
+concept of open and closed subsets on the real number line (such as `$(0,1)$`:m: and
+`$[1,2]$`:m:) are generalized to arbitrary sets.
 
-Formally, a *topology* on a set `$A$`:latex: is a collection `$\mathcal{T}$`:latex: of 
-subsets of `$A$`:latex: fufiling the criteria:
+Formally, a *topology* on a set `$A$`:m: is a collection `$\mathcal{T}$`:m: of 
+subsets of `$A$`:m: fufiling the criteria:
 
- 1. The empty set and the entire set `$A$`:latex: are both in `$\mathcal{T}$`:latex:.
+ 1. The empty set and the entire set `$A$`:m: are both in `$\mathcal{T}$`:m:.
 
- 2. The union of an arbitrary number of elements of `$\mathcal{T}$`:latex: is 
-    also in `$\mathcal{T}$`:latex:.
+ 2. The union of an arbitrary number of elements of `$\mathcal{T}$`:m: is 
+    also in `$\mathcal{T}$`:m:.
 
- 3. The intersection of a finite number of elements of `$\mathcal{T}$`:latex: is
-    also in `$\mathcal{T}$`:latex:.
+ 3. The intersection of a finite number of elements of `$\mathcal{T}$`:m: is
+    also in `$\mathcal{T}$`:m:.
 
-If a subset `$B$`:latex: of `$A$`:latex: is a member of `$\mathcal{T}$`:latex: then
-`$B$`:latex: is an open set under the topology `$\mathcal{T}$`:latex:.
+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:.
 
 *Coarseness* and *Fineness* are ways of comparing two topologies on the same space.
-`$\mathcal{T'}$`:latex: is finer than `$\mathcal{T}$`:latex: if `$\mathcal{T}$`:latex:
-is a subset of `$\mathcal{T'}$`:latex: (and `$\mathcal{T}$`:latex: is coarser); 
-it is *strictly finer* if it is a proper subset (and `$\mathcal{T}$`:latex: is 
-*strictly coarser*). Two sets are *comprable* if either `$\mathcal{T\in T'}$`:latex:
-or `$\mathcal{T'\in T}$`:latex:.
+`$\mathcal{T'}$`: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:.
 *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 
-    `$$(a,b)=\{x|a<x<b\}$$`:latex:
+    `$$(a,b)=\{x|a<x<b\}$$`:m:
     This is the usual definition for open sets on the real line.
 
 *Discrete Topology*
-    The topology on a set `$A$`:latex: consisting of all points of `$A$`:latex:;
-    in other words the power set of `$A$`:latex:.
+    The topology on a set `$A$`:m: consisting of all points of `$A$`:m:;
+    in other words the power set of `$A$`:m:.
 
 *Trivial/Indiscrete Topology*
-    The topology on a set `$A$`:latex: consisting of only the empty set and `$A$`:latex:
-    itself. Not super interesting but it's always there (when `$A$`:latex: isn't empty).
+    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).
     
-*Finite Complement Topology* (`$\mathcal{T_f}$`:latex:)
-    The topology on a set `$A$`:latex: consisting of the empty set any subset 
-    `$U$`:latex: such that `$A-U$`:latex: has a finite number of elements.
+*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.
 
-*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:latex:)
+*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:m:)
     The lower limit topology on the real line is generated by the collection of all half open
     intervals
-    `$$[a,b)=\{x|a\leq x<b\}$$`:latex:
-    `$\mathbb{R}_\mathcal{l}$`:latex: is strictly finer than the standard topology and
-    is not comprable to `$\mathbb{R}_K$`:latex:.
+    `$$[a,b)=\{x|a\leq x<b\}$$`:m:
+    `$\mathbb{R}_\mathcal{l}$`:m: is strictly finer than the standard topology and
+    is not comprable to `$\mathbb{R}_K$`:m:.
 
-*K-Topology* (`$\mathbb{R}_K$`:latex:)
+*K-Topology* (`$\mathbb{R}_K$`:m:)
     Let `$K$`:m: denote the set of all numbers `$1/n$`:n: where `$n$`:m: is
     a positive integer. 
     The K-topology on the real line is generated by the collection of all standard open intervals 
     minus `$K$`:m:. 
-    `$\mathbb{R}_K$`:latex: is strictly finer than the standard topology and is not
-    comprable to `$\mathbb{R}_\mathcal{l}$`:latex:.
+    `$\mathbb{R}_K$`:m: is strictly finer than the standard topology and is not
+    comprable to `$\mathbb{R}_\mathcal{l}$`:m:.
 
 *Order Topology*
     TODO
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