==================== 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 :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:`$A$`:m: are both in :m:`$\mathcal{T}$`. 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is also in `$\mathcal{T}$`:m:. 3. The intersection of a finite number of elements of `$\mathcal{T}$`:m: is also in `$\mathcal{T}$`:m:. If a subset `$B$`:m: of `$A$`:m: is a member of `$\mathcal{T}$`:m: then `$B$`:m: is an open set under the topology `$\mathcal{T}$`:m:. *Coarseness* and *Fineness* are ways of comparing two topologies on the same space. `$\mathcal{T'}$`:m: is finer than `$\mathcal{T}$`:m: if `$\mathcal{T}$`:m: is a subset of `$\mathcal{T'}$`:m: (and `$\mathcal{T}$`:m: is coarser); it is *strictly finer* if it is a proper subset (and `$\mathcal{T}$`:m: is *strictly coarser*). Two sets are *comprable* if either `$\mathcal{T\in T'}$`:m: or `$\mathcal{T'\in T}$`:m:. *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