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