**diff options**

Diffstat (limited to 'math')

-rw-r--r-- | math/topology | 14 |

1 files changed, 8 insertions, 6 deletions

diff --git a/math/topology b/math/topology index 79beeae..6f03eee 100644 --- a/math/topology +++ b/math/topology @@ -2,7 +2,7 @@ Topology ==================== -.. note:: Incomplete; in progress +.. warning:: Incomplete; in progress .. note:: Most of the definitions and notation in the section are based on [munkres]_ @@ -14,7 +14,7 @@ concept of open and closed subsets on the real number line (such as :m:`$(0,1)$` 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:`$\mathcal{T}$`. + 1. The empty set and the entire set :m:`$A$` are both in :m:`$\mathcal{T}$`. 2. The union of an arbitrary number of elements of :m:`$\mathcal{T}$` is also in :m:`$\mathcal{T}$`. @@ -34,6 +34,7 @@ or :m:`$\mathcal{T'\in T}$`. *Smaller* and *larger* are somtimes used instead of finer and coarser. Topologies can be generated from a *basis*. + TODO: Hausdorf Frequently Used Topologies @@ -42,6 +43,7 @@ Frequently Used Topologies *Standard Topology* The standard topology on the real line is generated by the collection of all intervals :m:`$$(a,b)=\{x|a<x<b\}$$` + This is the usual definition for open sets on the real line. *Discrete Topology* @@ -52,15 +54,15 @@ Frequently Used Topologies 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* (:m:`$\mathcal{T_f}$`) +*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* (:m:`$\mathbb{R}_\mathcal{l}$`) +*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 :m:`$$[a,b)=\{x|a\leq x<b\}$$` - :m:`$\mathbb{R}_\mathcal{l}$` is strictly finer than the standard topology and + :m:`$\mathbb{R}_{\mathcal{l}}$` is strictly finer than the standard topology and is not comprable to :m:`$\mathbb{R}_K$`. *K-Topology* (:m:`$\mathbb{R}_K$`) @@ -69,7 +71,7 @@ Frequently Used Topologies The K-topology on the real line is generated by the collection of all standard open intervals minus :m:`$K$`. :m:`$\mathbb{R}_K$` is strictly finer than the standard topology and is not - comprable to :m:`$\mathbb{R}_\mathcal{l}$`. + comprable to :m:`$\mathbb{R}_{\mathcal{l}}$`. *Order Topology* TODO |