From a290d583ca3c4dfc39115068f209d64449c93a03 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Sun, 24 Jan 2010 03:39:25 -0500 Subject: math fixes --- math/topology.page | 73 ++++++++++++++++++++++++++---------------------------- 1 file changed, 35 insertions(+), 38 deletions(-) (limited to 'math/topology.page') diff --git a/math/topology.page b/math/topology.page index 6f03eee..ea369fb 100644 --- a/math/topology.page +++ b/math/topology.page @@ -1,36 +1,33 @@ -==================== Topology ==================== -.. warning:: Incomplete; in progress - -.. note:: Most of the definitions and notation in the section are based on [munkres]_ +*References: 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 :m:$(0,1)$ and -:m:$[1,2]$) are generalized to arbitrary sets. +concept of open and closed subsets on the real number line (such as $(0,1)$ and +$[1,2]$) are generalized to arbitrary sets. -Formally, a *topology* on a set :m:$A$ is a collection :m:$\mathcal{T}$ of -subsets of :m:$A$ fufiling the criteria: +Formally, a *topology* on a set $A$ is a collection $\mathcal{T}$ of +subsets of $A$ fufiling the criteria: - 1. The empty set and the entire set :m:$A$ are both in :m:$\mathcal{T}$. + 1. The empty set and the entire set $A$ are both in $\mathcal{T}$. - 2. The union of an arbitrary number of elements of :m:$\mathcal{T}$ is - also in :m:$\mathcal{T}$. + 2. The union of an arbitrary number of elements of $\mathcal{T}$ is + also in $\mathcal{T}$. - 3. The intersection of a finite number of elements of :m:$\mathcal{T}$ is - also in :m:$\mathcal{T}$. + 3. The intersection of a finite number of elements of $\mathcal{T}$ is + also in $\mathcal{T}$. -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:$\mathcal{T}$. +If a subset $B$ of $A$ is a member of $\mathcal{T}$ then +$B$ is an open set under the topology $\mathcal{T}$. *Coarseness* and *Fineness* are ways of comparing two topologies on the same space. -: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:$\mathcal{T'\in T}$. +$\mathcal{T'}$ is finer than $\mathcal{T}$ if $\mathcal{T}$ +is a subset of $\mathcal{T'}$ (and $\mathcal{T}$ is coarser); +it is *strictly finer* if it is a proper subset (and $\mathcal{T}$ is +*strictly coarser*). Two sets are *comprable* if either $\mathcal{T\in T'}$ +or $\mathcal{T'\in T}$. *Smaller* and *larger* are somtimes used instead of finer and coarser. Topologies can be generated from a *basis*. @@ -42,40 +39,40 @@ 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