From a2d7b8c246db71e6c06e9f3db267b5a98691228d Mon Sep 17 00:00:00 2001
From: bnewbold
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