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authorbnewbold <bnewbold@eta.mit.edu>2008-11-05 02:18:57 -0500
committerbnewbold <bnewbold@eta.mit.edu>2008-11-05 02:18:57 -0500
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fixed latex math?
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@@ -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