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| -rw-r--r-- | math/topology | 62 |
1 files changed, 31 insertions, 31 deletions
diff --git a/math/topology b/math/topology index 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 - `$$(a,b)=\{x|a<x<b\}$$`:latex: + `$$(a,b)=\{x|a<x<b\}$$`:m: This is the usual definition for open sets on the real line. *Discrete Topology* - The topology on a set `$A$`:latex: consisting of all points of `$A$`:latex:; - in other words the power set of `$A$`:latex:. + The topology on a set `$A$`:m: consisting of all points of `$A$`:m:; + in other words the power set of `$A$`:m:. *Trivial/Indiscrete Topology* - The topology on a set `$A$`:latex: consisting of only the empty set and `$A$`:latex: - itself. Not super interesting but it's always there (when `$A$`:latex: isn't empty). + The topology on a set `$A$`:m: consisting of only the empty set and `$A$`:m: + itself. Not super interesting but it's always there (when `$A$`:m: isn't empty). -*Finite Complement Topology* (`$\mathcal{T_f}$`:latex:) - The topology on a set `$A$`:latex: consisting of the empty set any subset - `$U$`:latex: such that `$A-U$`:latex: has a finite number of elements. +*Finite Complement Topology* (`$\mathcal{T_f}$`:m:) + The topology on a set `$A$`:m: consisting of the empty set any subset + `$U$`:m: such that `$A-U$`:m: has a finite number of elements. -*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:latex:) +*Lower Limit Topology* (`$\mathbb{R}_\mathcal{l}$`:m:) The lower limit topology on the real line is generated by the collection of all half open intervals - `$$[a,b)=\{x|a\leq x<b\}$$`:latex: - `$\mathbb{R}_\mathcal{l}$`:latex: is strictly finer than the standard topology and - is not comprable to `$\mathbb{R}_K$`:latex:. + `$$[a,b)=\{x|a\leq x<b\}$$`:m: + `$\mathbb{R}_\mathcal{l}$`:m: is strictly finer than the standard topology and + is not comprable to `$\mathbb{R}_K$`:m:. -*K-Topology* (`$\mathbb{R}_K$`:latex:) +*K-Topology* (`$\mathbb{R}_K$`:m:) Let `$K$`:m: denote the set of all numbers `$1/n$`:n: where `$n$`:m: is a positive integer. The K-topology on the real line is generated by the collection of all standard open intervals minus `$K$`:m:. - `$\mathbb{R}_K$`:latex: is strictly finer than the standard topology and is not - comprable to `$\mathbb{R}_\mathcal{l}$`:latex:. + `$\mathbb{R}_K$`:m: is strictly finer than the standard topology and is not + comprable to `$\mathbb{R}_\mathcal{l}$`:m:. *Order Topology* TODO |