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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:`$$(a,b)=\{x|a<x<b\}$$` + $$(a,b)=\{x|a<x<b\}$$ This is the usual definition for open sets on the real line. *Discrete Topology* - The topology on a set :m:`$A$` consisting of all points of :m:`$A$`; - in other words the power set of :m:`$A$`. + The topology on a set $A$ consisting of all points of $A$; + in other words the power set of $A$. *Trivial/Indiscrete Topology* - The topology on a set :m:`$A$` consisting of only the empty set and :m:`$A$` - itself. Not super interesting but it's always there (when :m:`$A$` isn't empty). + The topology on a set $A$ consisting of only the empty set and $A$ + itself. Not super interesting but it's always there (when $A$ isn't empty). -*Finite Complement Topology* (:m:`$\mathcal{T}_f$`) - The topology on a set :m:`$A$` consisting of the empty set any subset - :m:`$U$` such that :m:`$A-U$` has a finite number of elements. +*Finite Complement Topology* ($\mathcal{T}_f$) + The topology on a set $A$ consisting of the empty set any subset + $U$ such that $A-U$ has a finite number of elements. -*Lower Limit Topology* (:m:`$\mathbb{R}_{\mathcal{l}}$`) +*Lower Limit Topology* ($\mathbb{R}_{\mathcal{l}}$) The lower limit topology on the real line is generated by the collection of all half open intervals - :m:`$$[a,b)=\{x|a\leq x<b\}$$` - :m:`$\mathbb{R}_{\mathcal{l}}$` is strictly finer than the standard topology and - is not comprable to :m:`$\mathbb{R}_K$`. + $$[a,b)=\{x|a\leq x<b\}$$ + $\mathbb{R}_{\mathcal{l}}$ is strictly finer than the standard topology and + is not comprable to $\mathbb{R}_K$. -*K-Topology* (:m:`$\mathbb{R}_K$`) - Let :m:`$K$` denote the set of all numbers :m:`$1/n$` where :m:`$n$` is +*K-Topology* ($\mathbb{R}_K$) + Let $K$ denote the set of all numbers $1/n$ where $n$ is a positive integer. The K-topology on the real line is generated by the collection of all standard open intervals - minus :m:`$K$`. - :m:`$\mathbb{R}_K$` is strictly finer than the standard topology and is not - comprable to :m:`$\mathbb{R}_{\mathcal{l}}$`. + minus $K$. + $\mathbb{R}_K$ is strictly finer than the standard topology and is not + comprable to $\mathbb{R}_{\mathcal{l}}$. *Order Topology* TODO -.. [munkres] `Topology (2nd edition)`:title:, by James R. Munkres. +[^munkres] **Topology (2nd edition)**, by James R. Munkres. |