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author | User <bnewbold@daemon.robocracy.org> | 2009-10-13 02:52:09 +0000 |
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committer | User <bnewbold@daemon.robocracy.org> | 2009-10-13 02:52:09 +0000 |
commit | f61026119df4700f69eb73e95620bc5928ca0fcb (patch) | |
tree | f17127cff9fec40f4207d9fa449b9692644ce6db /math/topology | |
parent | 9d431740a3e6a7caa09a57504856b5d1a4710a14 (diff) | |
download | knowledge-f61026119df4700f69eb73e95620bc5928ca0fcb.tar.gz knowledge-f61026119df4700f69eb73e95620bc5928ca0fcb.zip |
Grand rename for gitit transfer
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diff --git a/math/topology b/math/topology deleted file mode 100644 index 6f03eee..0000000 --- a/math/topology +++ /dev/null @@ -1,81 +0,0 @@ -==================== -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. |