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-rw-r--r--math/topology.page37
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.