From 930e322b9d0959b5a2067d9893b9d9ad92e64f56 Mon Sep 17 00:00:00 2001
From: bnewbold <bnewbold@eta.mit.edu>
Date: Tue, 4 Nov 2008 00:17:39 -0500
Subject: starting topology item

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+====================
+Topology
+====================
+
+.. note:: Incomplete; in progress
+
+.. note:: 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 `$(0,1)$`:latex: and
+`$[1,2]$`:latex:) 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:
+
+ 1. The empty set and the entire set `$A$`:latex: are both in `$\mathcal{T}$`:latex:.
+
+ 2. The union of an arbitrary number of elements of `$\mathcal{T}$`:latex: is 
+    also in `$\mathcal{T}$`:latex:.
+
+ 3. The intersection of a finite number of elements of `$\mathcal{T}$`:latex: is
+    also in `$\mathcal{T}$`:latex:.
+
+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:.
+
+*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:.
+*Smaller* and *larger* are somtimes used instead of finer and coarser.
+
+Topologies can be generated from a *basis*. 
+TODO: Hausdorf
+
+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:
+    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:.
+
+*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).
+    
+*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.
+
+*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:latex:)
+    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:.
+
+*K-Topology* (`$\mathbb{R}_K$`:latex:)
+    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:.
+
+*Order Topology*
+    TODO
+
+
+
+.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.
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