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====================
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 `$(0,1)$`:latex: and
`$[1,2]$`:latex:) are generalized to arbitrary sets.
Formally, a *topology* on a set `$A$`:latex: is a collection `$\mathcal{T}$`:latex: of
subsets of `$A$`:latex: fufiling the criteria:
1. The empty set and the entire set `$A$`:latex: are both in `$\mathcal{T}$`:latex:.
2. The union of an arbitrary number of elements of `$\mathcal{T}$`:latex: is
also in `$\mathcal{T}$`:latex:.
3. The intersection of a finite number of elements of `$\mathcal{T}$`:latex: is
also in `$\mathcal{T}$`:latex:.
If a subset `$B$`:latex: of `$A$`:latex: is a member of `$\mathcal{T}$`:latex: then
`$B$`:latex: is an open set under the topology `$\mathcal{T}$`:latex:.
*Coarseness* and *Fineness* are ways of comparing two topologies on the same space.
`$\mathcal{T'}$`:latex: is finer than `$\mathcal{T}$`:latex: if `$\mathcal{T}$`:latex:
is a subset of `$\mathcal{T'}$`:latex: (and `$\mathcal{T}$`:latex: is coarser);
it is *strictly finer* if it is a proper subset (and `$\mathcal{T}$`:latex: is
*strictly coarser*). Two sets are *comprable* if either `$\mathcal{T\in T'}$`:latex:
or `$\mathcal{T'\in T}$`:latex:.
*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
`$$(a,b)=\{x|a<x<b\}$$`:latex:
This is the usual definition for open sets on the real line.
*Discrete Topology*
The topology on a set `$A$`:latex: consisting of all points of `$A$`:latex:;
in other words the power set of `$A$`:latex:.
*Trivial/Indiscrete Topology*
The topology on a set `$A$`:latex: consisting of only the empty set and `$A$`:latex:
itself. Not super interesting but it's always there (when `$A$`:latex: isn't empty).
*Finite Complement Topology* (`$\mathcal{T_f}$`:latex:)
The topology on a set `$A$`:latex: consisting of the empty set any subset
`$U$`:latex: such that `$A-U$`:latex: has a finite number of elements.
*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:latex:)
The lower limit topology on the real line is generated by the collection of all half open
intervals
`$$[a,b)=\{x|a\leq x<b\}$$`:latex:
`$\mathbb{R}_\mathcal{l}$`:latex: is strictly finer than the standard topology and
is not comprable to `$\mathbb{R}_K$`:latex:.
*K-Topology* (`$\mathbb{R}_K$`:latex:)
Let `$K$`:m: denote the set of all numbers `$1/n$`:n: where `$n$`:m: is
a positive integer.
The K-topology on the real line is generated by the collection of all standard open intervals
minus `$K$`:m:.
`$\mathbb{R}_K$`:latex: is strictly finer than the standard topology and is not
comprable to `$\mathbb{R}_\mathcal{l}$`:latex:.
*Order Topology*
TODO
.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.
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