neighborhood system on a set
However, one can start out with the definition of a “abstract neighborhood system” on an arbitrary set and define a topology on based on this system so that is the neighborhood system of . This is done as follows:
Let be a set and be a subset of , where is the power set of . Then is said to be a abstract neighborhood system of if the following conditions are satisfied:
if , then ,
for every , there is a such that ,
if and , then ,
if , then ,
if , then there is a such that
for all .
In addition, given this , define the abstract neighborhood system around to be the subset of consisting of all those elements whose first coordinate is . Evidently, is the disjoint union of for all . Finally, let
The two definitions are the same by condition 3. We assert that defined above is a topology on . Furthermore, is the set of neighborhoods of under .
We first show that is a topology. For every , some , we have by condition 2. Hence by condition 3. So . Also, is vacuously satisfied, for no . If , then by condition 4. Let be a subset of whose elements are indexed by (). Let . Pick any , then for some . Since , . Since , by condition 3, so .
Next, suppose is the set of neighborhoods of under . We need to show :
(). If , then there is with . But , so by condition 3, , or , or .
(). Pick any and set . Then by condition 1. We show is open. This means we need to find, for each , a such that . If , then . By condition 5, there is such that , and for any , , or by condition 1. So by the definition of , or . Thus is open and .
(condition 1): clear
(condition 2): because for each
(condition 3): if is a neighborhood of and a supserset of , then is also a neighborhood of
(condition 4): if and are neighborhoods of , there are open with and , so , which means is a neighborhood of
(condition 5): if is a neighborhood of , there is open with ; clearly is a neighborhood of and any has as neighborhood.
So the definition of a neighborhood system on an arbitrary set gives an alternative way of defining a topology on the set. There is a one-to-one correspondence between the set of topologies on a set and the set of abstract neighborhood systems on the set.
|Title||neighborhood system on a set|
|Date of creation||2013-03-22 16:41:34|
|Last modified on||2013-03-22 16:41:34|
|Last modified by||CWoo (3771)|
|Defines||abstract neighborhood system|