summaryrefslogtreecommitdiffstats
path: root/math/topology.page
diff options
context:
space:
mode:
authorUser <bnewbold@daemon.robocracy.org>2009-10-13 02:52:09 +0000
committerUser <bnewbold@daemon.robocracy.org>2009-10-13 02:52:09 +0000
commitf61026119df4700f69eb73e95620bc5928ca0fcb (patch)
treef17127cff9fec40f4207d9fa449b9692644ce6db /math/topology.page
parent9d431740a3e6a7caa09a57504856b5d1a4710a14 (diff)
downloadknowledge-f61026119df4700f69eb73e95620bc5928ca0fcb.tar.gz
knowledge-f61026119df4700f69eb73e95620bc5928ca0fcb.zip
Grand rename for gitit transfer
Diffstat (limited to 'math/topology.page')
-rw-r--r--math/topology.page81
1 files changed, 81 insertions, 0 deletions
diff --git a/math/topology.page b/math/topology.page
new file mode 100644
index 0000000..6f03eee
--- /dev/null
+++ b/math/topology.page
@@ -0,0 +1,81 @@
+====================
+Topology
+====================
+
+.. warning:: 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 :m:`$(0,1)$` and
+:m:`$[1,2]$`) are generalized to arbitrary sets.
+
+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:`$A$` are both in :m:`$\mathcal{T}$`.
+
+ 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is
+ also in :m:`$\mathcal{T}$`.
+
+ 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is
+ also in :m:`$\mathcal{T}$`.
+
+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.
+: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*.
+
+TODO: Hausdorf
+
+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*
+ 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 :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:`$\mathcal{T}_f$`)
+ 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:`$\mathbb{R}_{\mathcal{l}}$`)
+ 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
+ is not comprable to :m:`$\mathbb{R}_K$`.
+
+*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 :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
+
+
+
+.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.