From 0b113afaa8ae15d3672c6c50f6f6b5b26d78d618 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Wed, 5 Nov 2008 02:26:39 -0500 Subject: better for now --- math/topology | 54 +++++++++++++++++++++++++++--------------------------- 1 file changed, 27 insertions(+), 27 deletions(-) (limited to 'math') 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