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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 |

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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. |