Topology ==================== *References: 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 $(0,1)$ and $[1,2]$) are generalized to arbitrary sets. Formally, a *topology* on a set $A$ is a collection $\mathcal{T}$ of subsets of $A$ fufiling the criteria: 1. The empty set and the entire set $A$ are both in $\mathcal{T}$. 2. The union of an arbitrary number of elements of $\mathcal{T}$ is also in $\mathcal{T}$. 3. The intersection of a finite number of elements of $\mathcal{T}$ is also in $\mathcal{T}$. If a subset $B$ of $A$ is a member of $\mathcal{T}$ then $B$ is an open set under the topology $\mathcal{T}$. *Coarseness* and *Fineness* are ways of comparing two topologies on the same space. $\mathcal{T'}$ is finer than $\mathcal{T}$ if $\mathcal{T}$ is a subset of $\mathcal{T'}$ (and $\mathcal{T}$ is coarser); it is *strictly finer* if it is a proper subset (and $\mathcal{T}$ is *strictly coarser*). Two sets are *comprable* if either $\mathcal{T \in T'}$ or $\mathcal{T' \in T}$. *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 (a,b)=\{x|a