==================== 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