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-rw-r--r-- | math/topology | 54 |

1 files changed, 27 insertions, 27 deletions

diff --git a/math/topology b/math/topology index 6b44484..79beeae 100644 --- a/math/topology +++ b/math/topology @@ -17,20 +17,20 @@ 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:. + also in :m:`$\mathcal{T}$`. - 3. The intersection of a finite number of elements of `$\mathcal{T}$`:m: is - also in `$\mathcal{T}$`:m:. + 3. The intersection of a finite number of elements of :m:`$\mathcal{T}$` is + also in :m:`$\mathcal{T}$`. -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:. +If a subset :m:`$B$` of :m:`$A$` is a member of :m:`$\mathcal{T}$` then +:m:`$B$` is an open set under the topology :m:`$\mathcal{T}$`. *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:. +:m:`$\mathcal{T'}$` is finer than :m:`$\mathcal{T}$` if :m:`$\mathcal{T}$` +is a subset of :m:`$\mathcal{T'}$` (and :m:`$\mathcal{T}$` is coarser); +it is *strictly finer* if it is a proper subset (and :m:`$\mathcal{T}$` is +*strictly coarser*). Two sets are *comprable* if either :m:`$\mathcal{T\in T'}$` +or :m:`$\mathcal{T'\in T}$`. *Smaller* and *larger* are somtimes used instead of finer and coarser. Topologies can be generated from a *basis*. @@ -41,35 +41,35 @@ Frequently Used Topologies *Standard Topology* The standard topology on the real line is generated by the collection of all intervals - `$$(a,b)=\{x|a<x<b\}$$`:m: + :m:`$$(a,b)=\{x|a<x<b\}$$` 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:. + The topology on a set :m:`$A$` consisting of all points of :m:`$A$`; + in other words the power set of :m:`$A$`. *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). + The topology on a set :m:`$A$` consisting of only the empty set and :m:`$A$` + itself. Not super interesting but it's always there (when :m:`$A$` 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 `$A-U$`:m: has a finite number of elements. +*Finite Complement Topology* (:m:`$\mathcal{T_f}$`) + The topology on a set :m:`$A$` consisting of the empty set any subset + :m:`$U$` such that :m:`$A-U$` has a finite number of elements. -*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:m:) +*Lower Limit Topology* (:m:`$\mathbb{R}_\mathcal{l}$`) The lower limit topology on the real line is generated by the collection of all half open intervals - `$$[a,b)=\{x|a\leq x<b\}$$`:m: - `$\mathbb{R}_\mathcal{l}$`:m: is strictly finer than the standard topology and - is not comprable to `$\mathbb{R}_K$`:m:. + :m:`$$[a,b)=\{x|a\leq x<b\}$$` + :m:`$\mathbb{R}_\mathcal{l}$` is strictly finer than the standard topology and + is not comprable to :m:`$\mathbb{R}_K$`. -*K-Topology* (`$\mathbb{R}_K$`:m:) - Let `$K$`:m: denote the set of all numbers `$1/n$`:n: where `$n$`:m: is +*K-Topology* (:m:`$\mathbb{R}_K$`) + Let :m:`$K$` denote the set of all numbers :m:`$1/n$` where :m:`$n$` is a positive integer. The K-topology on the real line is generated by the collection of all standard open intervals - minus `$K$`:m:. - `$\mathbb{R}_K$`:m: is strictly finer than the standard topology and is not - comprable to `$\mathbb{R}_\mathcal{l}$`:m:. + minus :m:`$K$`. + :m:`$\mathbb{R}_K$` is strictly finer than the standard topology and is not + comprable to :m:`$\mathbb{R}_\mathcal{l}$`. *Order Topology* TODO |