Thymidylate kinase

2011-07-05T14:06:48Z

84.203.168.85: Grammar only: Mild misuse of "an".

In [[general topology]], a branch of mathematics, the '''Appert topology''', named for {{harvtxt|Appert|1934}}, is an example of a [[topology]] on the set {{nowrap|1='''Z'''+ = {1, 2, 3, …}}} of [[natural number|positive integers]].<ref name="CEIT">{{harvnb|Steen|Seebach|1995|pp=117–118}}</ref> To give '''Z'''+ a topology means to say which [[subset]]s of '''Z'''+ are [[open set|open]] in a manner that satisfies certain axioms:<ref>{{harvnb|Steen|Seebach|1995|p=3}}</ref>

# The [[union (mathematics)|union]] of open sets is an open set.
# The finite [[intersection (mathematics)|intersection]] of open sets is an open set.
# '''Z'''+ and the [[empty set]] ∅ are open sets.

In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer.

== Construction ==

Let ''S'' be a subset of '''Z'''+, and let {{nowrap|1=N(''n'',''S'')}} denote the number of elements of ''S'' which are less than or equal to ''n'':
:<math> \mathrm{N}(n,S) = \#\{ m \in S : m \le n \} . </math>
In Appert's topology, a set ''S'' is defined to be open if either it does not contain 1 or N(''n'',''S'')/''n'' tends towards 1 as ''n'' tends towards infinity:<ref name="CEIT"/>
:<math>\lim_{n \to \infty} \frac{\text{N}(n,S)}{n} = 1.</math>
The empty set is an open set in this topology because ∅ is a set that does not contain 1, and the whole set '''Z'''+ is also open in this topology since
:<math> \text{N}\!\left(n,{\bold Z}^+ \right) = n \ , </math>
meaning that {{nowrap|1=N(''n'',''S'')/''n'' = 1}} for all ''n''.

== Related topologies ==
The Appert topology is closely related to the [[Fort space]] topology that arises from giving the set of integers greater than one the [[discrete topology]], and then taking the point 1 as the point at infinity in a [[one point compactification]] of the space.<ref name="CEIT"/> The Fort space is a refinement of the Appert topology.

==Properties==
The closed subsets of '''Z'''+, equipped with the Appert topology, are the subsets ''S'' that either contain 1 or for which
:<math>\lim_{n\to\infty} \frac{\mathrm{N}(n,S)}{n} = 0.</math>
As a result, '''Z'''+ is a [[completely normal space]] (and thus also [[Hausdorff space|Hausdorff]]), for suppose that ''A'' and ''B'' are disjoint closed sets. If {{nowrap|''A'' ∪ ''B''}} did not contain 1, then ''A'' and ''B'' would also be open and thus completely separated. On the other hand, if ''A'' contains 1 then ''B'' is open and <math>\scriptstyle \lim_{n\to\infty} \mathrm{N}(n,B)/n \, = \, 0</math>, so that '''Z'''+−''B'' is an open neighborhood of ''A'' disjoint from ''B''.<ref name="CEIT"/>

A subset of '''Z'''+ is [[compact space|compact]] in the Appert topology if and only if it is finite. In particular, '''Z'''+ is not [[locally compact space|locally compact]], since there is no compact neighborhood of 1. Moreover, '''Z'''+ is not [[countably compact]].<ref name="CEIT"/>

== Notes ==

{{reflist}}

==References==
* {{Citation|first=Q|last=Appert|title=Propriétés des Espaces Abstraits les Ples Generaux|series=Actual. Sci. Ind.|number=146|publisher=Hermann|year=1934}}.
* {{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|publisher=Dover|year=1995|ISBN=0-486-68735-X}}.

[[Category:General topology]]
[[Category:Topological spaces]]
[[Category:Topologies on the set of positive integers]]

formulasearchengine - User contributions [en]

Thymidylate kinase