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In mathematics, more specifically [[general topology]], the '''divisor topology''' is an example of a [[topology]] given to the set ''X'' of positive [[integer]]s that are greater than or equal to two, i.e., {{nowrap|1=''X'' = {2, 3, 4, 5, …}}}. The divisor topology is the [[poset topology]] for the [[partial order]] relation of [[divisibility]] on the positive integers.
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To give the set ''X'' a topology means to say which [[subset]]s of ''X'' are "open", and to do so in a way that the following [[axiom]]s are met:<ref name="CEIT">{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|publisher=Dover|year=1995|ISBN=0-486-68735-X}}</ref>
 
# The [[union (mathematics)|union]] of open sets is an open set.
# The finite [[intersection (mathematics)|intersection]] of open sets is an open set.
# The set ''X'' and the [[empty set]] ∅ are open sets.
 
== Construction ==
 
The set ''X'' and the empty set ∅ are required to be open sets, and so we define ''X'' and ∅ to be open sets in this topology. Denote by '''Z'''<sup>+</sup> the set of positive [[integer]]s, i.e., the set of positive whole number greater than or equal to one. Read the notation ''x''|''n'' as "''x'' divides ''n''", and consider the sets
:<math>S_n = \{x \in \bold{Z}^+ : x|n \} </math>  
Then the set ''S<sub>n</sub>'' is the set of [[divisor]]s of ''n''. For different values of ''n'', the sets ''S<sub>n</sub>'' are used as a [[basis (topology)|basis]] for the divisor topology.<ref name="CEIT"/>
 
The open sets in this topology are the [[lower set]]s for the partial order defined by {{nowrap|1=''x'' ≤ ''y'' if ''x''&thinsp;{{!}}&thinsp;''y''.}}
 
== Properties ==
* The set of [[prime number]]s is [[dense (topology)|dense]] in ''X''. In fact, every dense open set must include every prime, and therefore ''X'' is a [[Baire space]].<ref name="CEIT"/>
* ''X'' is a [[Kolmogorov space]] that is not [[T1 space|T1]]. In particular, it is [[Hausdorff space|non-Hausdorff]].
* ''X'' is [[second countable space|second countable]].
* ''X'' is [[connected space|connected]] and [[locally connected]].
* ''X'' is not [[compact space|compact]], since the basic open sets ''S''<sub>''n''</sub> comprise an infinite covering with no finite subcovering. ''X'' is not [[locally compact]].
* The closure of a point in ''x'' is the set of all multiples of ''x''.
 
==See also==
* [[Zariski topology]]: A topology on the integers whose open sets are the complements of [[prime ideal]]s.
 
== References ==
{{Reflist}}
 
 
{{DEFAULTSORT:Divisor topology}}
[[Category:General topology]]
[[Category:Topologies on the set of positive integers]]

Latest revision as of 16:42, 18 June 2014

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One pimple can instantly wreck your day. When a blemish seems in your in any other case flawless pores and skin, it may be an indication that more pimples are on the way. When you've got greasy skin, you are particularly liable to get pimples due to the buildup of bacteria. There are lots of business pimple cures that promise to keep the ugly bumps at bay. However http://www.medicinenet.com/acne/patient-comments-492.htm, these products may be pricey and should contain components that may irritate your skin. To save cash, make your individual all-pure pimple-prevention facial masks at home. You may also like my article on Learn how to eliminate a zit quick HERE - and it's not by using toothpaste!

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Don’t hesitate to visit an expert! Over-the-counter acne treatments will be costly, and sometimes don't eliminate zits sufficiently. Even very delicate pimples is greatest treated with prescription medicines or in-workplace acne therapies. Your dermatologist can be able recommend the most effective remedy in your skin and eliminate zits as quickly as doable. You aren’t stuck together with your pimples scars! Although scarring is everlasting, if left untreated, there are excellent remedies now available for zits scars. We make the most of the Fractionated CO2 Laser , which stimulates your skin’s own collagen to fill within the pitted pimples scars. Most patients require just one therapy.

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Stumped on how to get "Godzilla" off of my pals nostril, my mother walked in with a bottle of "Listerine Mouth Wash." She said pour some of this onto a washcloth and wash your face with it. It appeared idiotic but we trusted her. We left the Listerine on the surface for about 10 minutes earlier than washing it off. My good friend mentioned she felt one thing tingling in her pores. My mother instructed her to wash it once more the identical approach earlier than mattress time and within the morning. She also advisable her to go away toothpaste on the pimple in a single day.

Nobody factor causes acne. Zits occurs when sebaceous (oil) glands hooked up to the hair follicles are stimulated at the time of puberty by circulating male hormones. Sebum (oil) is a pure substance which lubricates and protects the pores and skin. Related to elevated oil manufacturing is a change within the manner through http://www.medicinenet.com/acne/article.htm which the skin cells mature, predisposing them to clog the follicular pore. The clog can seem as a whitehead if it coated by a skinny layer of skin or if exposed to the air the darker uncovered portion of the plug is named a "blackhead." The plugged hair follicle gradually enlarges, producing a bump.

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