From 930e322b9d0959b5a2067d9893b9d9ad92e64f56 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Tue, 4 Nov 2008 00:17:39 -0500 Subject: starting topology item --- math/topology | 79 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 79 insertions(+) create mode 100644 math/topology (limited to 'math') diff --git a/math/topology b/math/topology new file mode 100644 index 0000000..104cbe8 --- /dev/null +++ b/math/topology @@ -0,0 +1,79 @@ +==================== +Topology +==================== + +.. note:: Incomplete; in progress + +.. note:: 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 $(0,1)$:latex: and +$[1,2]$:latex:) are generalized to arbitrary sets. + +Formally, a *topology* on a set $A$:latex: is a collection $\mathcal{T}$:latex: of +subsets of $A$:latex: fufiling the criteria: + + 1. The empty set and the entire set $A$:latex: are both in $\mathcal{T}$:latex:. + + 2. The union of an arbitrary number of elements of $\mathcal{T}$:latex: is + also in $\mathcal{T}$:latex:. + + 3. The intersection of a finite number of elements of $\mathcal{T}$:latex: is + also in $\mathcal{T}$:latex:. + +If a subset $B$:latex: of $A$:latex: is a member of $\mathcal{T}$:latex: then +$B$:latex: is an open set under the topology $\mathcal{T}$:latex:. + +*Coarseness* and *Fineness* are ways of comparing two topologies on the same space. +$\mathcal{T'}$:latex: is finer than $\mathcal{T}$:latex: if $\mathcal{T}$:latex: +is a subset of $\mathcal{T'}$:latex: (and $\mathcal{T}$:latex: is coarser); +it is *strictly finer* if it is a proper subset (and $\mathcal{T}$:latex: is +*strictly coarser*). Two sets are *comprable* if either $\mathcal{T\in T'}$:latex: +or $\mathcal{T'\in T}$:latex:. +*Smaller* and *larger* are somtimes used instead of finer and coarser. + +Topologies can be generated from a *basis*. +TODO: Hausdorf + +Frequently Used Topologies +============================ + +*Standard Topology* + The standard topology on the real line is generated by the collection of all intervals + `(a,b)=\{x|a