Cofinite

In mathematics, a cofinite subset of a set X is a subset Y such that its complement in X is a finite set. In other words, Y contains all but finitely many elements of X

The cofinite topology on any set X makes X into a topological space such that the only closed subsets are finite sets, or the whole of X. Then X is automatically compact in this topology, since every open set only omits finitely many points of X. Put another way, the open sets in this topology are the cofinite sets, plus the empty set.

One place where this concept occurs naturally is in the context of the Zariski topology. Since polynomials over a field K are zero on finite sets, or the whole of K, the Zariski topology on K (considered as affine line) is the cofinite topology. The same is true for any irreducible algebraic curve; it is not true, for example, for XY = 0 in the plane.



In the News

UCSD Biologists Identify Gene In Corn Plants That May Have Paved Way
Biologists at the University of California, San Diego have identified a gene that appears to have been a critical trait in allowing the earliest plant breeders 7,000 years ago to transform teosinte, a wild grass that grows in the Mexican Sierra Madre, into maize, the world's third most planted crop after rice and wheat.

Resisting Peer Pressure: New Findings Shed Light On Adolescent Decisio
The capacity to resist peer pressure in early adolescence may depend on the strength of connections between certain areas of the brain, according to a new study. Findings indicate that brain regions which regulate different aspects of behaviour are more interconnected in children with high resistance to peer influence.

Australia's Megafauna Coexisted With Humans
Analyses of ancient fossils suggest that early Australian Aborigines did not wipe out the continent's megafauna in a frenzied hunting rampage. New research conducted by Australian and British scientists reveals that in fact humans and megafauna, such as gigantic three tonne wombat-like creatures, a ferocious marsupial "lion"and the world's all-time biggest lizard, may have co-existed for around 15,000 years.

Your Right to Be an Idiot
Bad information and outright lies pulled off the internet can cause harm. In a benign case, maybe you louse up a term paper. In its malignant form, you could ruina life. What, if anything, should be done about it? Commentary by Tony Long.

History of Gay and Lesbian Pride Month
Brief information about the reason for the June celebration of Gay and Lesbian Pride Month. The site notes that the celebration's origin can be traced to the June 1969 protests over a raid to enforce alcohol control laws at the "Stonewall Inn, a popular gay bar in New York City's Greenwich Village."From the National Women's History Project.

Mixing Large Doses Of Common Painkiller And Caffeine May Increase Risk
Consuming large amounts of caffeine while taking acetaminophen, a widely used painkiller, could potentially cause liver damage, according to a preliminary laboratory study. The toxic interaction could occur not only from drinking caffeinated beverages while taking the painkiller but also from using large amounts of medications that intentionally combine caffeine and acetaminophen, the researchers say.

'Nanoshells' Simultaneously Detect And Destroy Cancerous Cells
Researchers at Rice University in Texas have developed a new approach to fighting cancer, based on nanoscale particles that can both detect and destroy cancerous cells. The report appears in the April 13 issue of the American Chemical Society's journal Nano Letters. ACS is the world's largest scientific society.

Pre-Columbian Ruin Discovered In Peru
Explorer Keith Muscutt has announced the existence of a previously unknown pre-Columbian ruin in Peru -- The Huaca La Penitenciarķa de la Meseta, which will be featured in Discovery Channel's new series, CHASING MUMMIES, premiering January 2008.

E-Health Gaffe Exposes Hospital
An Indiana computer consultant finds a password hard-coded into a popular medical office application, and that leads to patient data from a hospital in Washington, D.C. By Kevin Poulsen.

Will IPhone Be Just Another IPod?
Forget pushing the envelope: Apple is likely to take it easy on its allegedly forthcoming phone, an analyst predicts. In Gear Factor.


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