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Diffstat (limited to 'math/topology.page')

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1 files changed, 17 insertions, 20 deletions

diff --git a/math/topology.page b/math/topology.page index ea369fb..9701427 100644 --- a/math/topology.page +++ b/math/topology.page @@ -26,8 +26,8 @@ $B$ is an open set under the topology $\mathcal{T}$. $\mathcal{T'}$ 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 $\mathcal{T'\in T}$. +*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*. @@ -37,42 +37,39 @@ TODO: Hausdorf Frequently Used Topologies ============================ -*Standard Topology* - The standard topology on the real line is generated by the collection of all intervals +Standard Topology +: The standard topology on the real line is generated by the collection of all intervals $$(a,b)=\{x|a<x<b\}$$ - This is the usual definition for open sets on the real line. -*Discrete Topology* - The topology on a set $A$ consisting of all points of $A$; +Discrete Topology +: 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 $A$ consisting of only the empty set and $A$ +Trivial/Indiscrete Topology +: 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* ($\mathcal{T}_f$) - The topology on a set $A$ consisting of the empty set any subset +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* ($\mathbb{R}_{\mathcal{l}}$) - The lower limit topology on the real line is generated by the collection of all half open +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 $$[a,b)=\{x|a\leq x<b\}$$ $\mathbb{R}_{\mathcal{l}}$ is strictly finer than the standard topology and is not comprable to $\mathbb{R}_K$. -*K-Topology* ($\mathbb{R}_K$) - Let $K$ denote the set of all numbers $1/n$ where $n$ 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 $K$. $\mathbb{R}_K$ is strictly finer than the standard topology and is not comprable to $\mathbb{R}_{\mathcal{l}}$. -*Order Topology* - TODO - - +Order Topology +: TODO -[^munkres] **Topology (2nd edition)**, by James R. Munkres. +[^munkres]: **Topology (2nd edition)**, by James R. Munkres. |