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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 :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
`$$(a,b)=\{xa<x<b\}$$`:m:
This is the usual definition for open sets on the real line.
*Discrete Topology*
The topology on a set `$A$`:m: consisting of all points of `$A$`:m:;
in other words the power set of `$A$`:m:.
*Trivial/Indiscrete Topology*
The topology on a set `$A$`:m: consisting of only the empty set and `$A$`:m:
itself. Not super interesting but it's always there (when `$A$`:m: isn't empty).
*Finite Complement Topology* (`$\mathcal{T_f}$`:m:)
The topology on a set `$A$`:m: consisting of the empty set any subset
`$U$`:m: such that `$AU$`:m: has a finite number of elements.
*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:m:)
The lower limit topology on the real line is generated by the collection of all half open
intervals
`$$[a,b)=\{xa\leq x<b\}$$`:m:
`$\mathbb{R}_\mathcal{l}$`:m: is strictly finer than the standard topology and
is not comprable to `$\mathbb{R}_K$`:m:.
*KTopology* (`$\mathbb{R}_K$`:m:)
Let `$K$`:m: denote the set of all numbers `$1/n$`:n: where `$n$`:m: is
a positive integer.
The Ktopology on the real line is generated by the collection of all standard open intervals
minus `$K$`:m:.
`$\mathbb{R}_K$`:m: is strictly finer than the standard topology and is not
comprable to `$\mathbb{R}_\mathcal{l}$`:m:.
*Order Topology*
TODO
.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres.
