Topology
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*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
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Standard Topology
: The standard topology on the real line is generated by the collection of all intervals
$$(a,b)=\{x|a