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diff --git a/math/topology b/math/topology new file mode 100644 index 0000000..104cbe8 --- /dev/null +++ b/math/topology @@ -0,0 +1,79 @@ +==================== +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. |