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 diff --git a/math/topology b/math/topologyindex 104cbe8..c7e482f 100644--- a/math/topology+++ b/math/topology@@ -8,29 +8,29 @@ Topology 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.+concept of open and closed subsets on the real number line (such as $(0,1)$:m: and+$[1,2]$:m:) 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:+Formally, a *topology* on a set $A$:m: is a collection $\mathcal{T}$:m: of +subsets of $A$:m: fufiling the criteria: - 1. The empty set and the entire set $A$:latex: are both in $\mathcal{T}$:latex:.+ 1. The empty set and the entire set $A$:m: are both in $\mathcal{T}$:m:. - 2. The union of an arbitrary number of elements of $\mathcal{T}$:latex: is - also in $\mathcal{T}$:latex:.+ 2. The union of an arbitrary number of elements of $\mathcal{T}$:m: is + also in $\mathcal{T}$:m:. - 3. The intersection of a finite number of elements of $\mathcal{T}$:latex: is- also in $\mathcal{T}$:latex:.+ 3. The intersection of a finite number of elements of $\mathcal{T}$:m: is+ also in $\mathcal{T}$:m:. -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:.+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:. *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:.+$\mathcal{T'}$: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:. *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