From 7fb2bedfc29bb6a52520f280ce73b7491e071740 Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@eta.mit.edu>
Date: Wed, 5 Nov 2008 02:30:02 -0500
Subject: better for now
---
math/topology | 14 ++++++++------
1 file changed, 8 insertions(+), 6 deletions(-)
(limited to 'math')
diff --git a/math/topology b/math/topology
index 79beeae..6f03eee 100644
--- a/math/topology
+++ b/math/topology
@@ -2,7 +2,7 @@
Topology
====================
-.. note:: Incomplete; in progress
+.. warning:: Incomplete; in progress
.. note:: Most of the definitions and notation in the section are based on [munkres]_
@@ -14,7 +14,7 @@ concept of open and closed subsets on the real number line (such as :m:`$(0,1)$`
Formally, a *topology* on a set :m:`$A$` is a collection :m:`$\mathcal{T}$` of
subsets of :m:`$A$` fufiling the criteria:
- 1. The empty set and the entire set :m:`$A$`:m: are both in :m:`$\mathcal{T}$`.
+ 1. The empty set and the entire set :m:`$A$` are both in :m:`$\mathcal{T}$`.
2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is
also in :m:`$\mathcal{T}$`.
@@ -34,6 +34,7 @@ or :m:`$\mathcal{T'\in T}$`.
*Smaller* and *larger* are somtimes used instead of finer and coarser.
Topologies can be generated from a *basis*.
+
TODO: Hausdorf
Frequently Used Topologies
@@ -42,6 +43,7 @@ Frequently Used Topologies
*Standard Topology*
The standard topology on the real line is generated by the collection of all intervals
:m:`$$(a,b)=\{x|a<x<b\}$$`
+
This is the usual definition for open sets on the real line.
*Discrete Topology*
@@ -52,15 +54,15 @@ Frequently Used Topologies
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* (:m:`$\mathcal{T_f}$`)
+*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* (:m:`$\mathbb{R}_\mathcal{l}$`)
+*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
:m:`$$[a,b)=\{x|a\leq x<b\}$$`
- :m:`$\mathbb{R}_\mathcal{l}$` is strictly finer than the standard topology and
+ :m:`$\mathbb{R}_{\mathcal{l}}$` is strictly finer than the standard topology and
is not comprable to :m:`$\mathbb{R}_K$`.
*K-Topology* (:m:`$\mathbb{R}_K$`)
@@ -69,7 +71,7 @@ Frequently Used Topologies
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:`$\mathbb{R}_\mathcal{l}$`.
+ comprable to :m:`$\mathbb{R}_{\mathcal{l}}$`.
*Order Topology*
TODO
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