From f61026119df4700f69eb73e95620bc5928ca0fcb Mon Sep 17 00:00:00 2001 From: User Date: Tue, 13 Oct 2009 02:52:09 +0000 Subject: Grand rename for gitit transfer --- math/topology.page | 81 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 81 insertions(+) create mode 100644 math/topology.page (limited to 'math/topology.page') 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