In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T₄: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T₄ space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T₅ spaces, and perfectly normal Hausdorff spaces, or T₆ spaces.

Definitions[link]

A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.

The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods U and V, here represented by larger, but still disjoint, open disks.

A T₄ space is a T₁ space X that is normal; this is equivalent to X being Hausdorff and normal.

A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.

A completely T₄ space, or T₅ space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T₄ space.

A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets E and F can be precisely separated by a continuous function f from X to the real line R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval [0,1].)

It turns out that X is perfectly normal if and only if X is normal and every closed set is a G_δ set. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal.^{[citation needed]}

A Hausdorff perfectly normal space X is a T₆ space, or perfectly T₄ space.

Note that the terms "normal space" and "T₄" and derived concepts occasionally have a different meaning. (Nonetheless, "T₅" always means the same as "completely T₄", whatever that may be.) The definitions given here are the ones usually used today, and the ones used in Wikipedia articles on separation axioms. For more on this issue, see History of the separation axioms.

Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature — they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T₄ space. Given the historical confusion of the meaning of the terms, we prefer verbal descriptions when applicable, that is, "normal Hausdorff" instead of "T₄", or "completely normal Hausdorff" instead of "T₅".

Fully normal spaces and fully T₄ spaces are discussed elsewhere; they are related to paracompactness.

A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.

Examples of normal spaces[link]

Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:

• All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
• All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
• All compact Hausdorff spaces are normal;
• In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;
• Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
• All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompact manifolds which are not even normal.
• All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
• Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space that is not regular.

Examples of non-normal spaces[link]

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.

A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that the product of uncountably many non-compact Hausdorff spaces is never normal.

Properties[link]

The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space X.

Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to be entirely within the unit interval [0,1]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.)

More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A to R, then there exists a continuous function F: X → R which extends f in the sense that F(x) = f(x) for all x in A.

If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U. (This shows the relationship of normal spaces to paracompactness.)

In fact, any space that satisfies any one of these conditions must be normal.

A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example of this phenomenon is the Sorgenfrey plane. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank.

Relationships to other separation axioms[link]

If a normal space is R₀, then it is in fact completely regular. Thus, anything from "normal R₀" to "normal completely regular" is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normal T₁ spaces are Tychonoff. These are what we normally call normal Hausdorff spaces.

Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.

References[link]

• Kemoto, Nobuyuki (2004). "Higher Separation Axioms". In K.P. Hart, J. Nagata, and J.E. Vaughan. Encyclopedia of General Topology. Amsterdam: Elsevier Science. ISBN 0-444-50355-2. 
• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6 (Dover edition). 

This biographical article needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (July 2010)

Danny Trejo
Danny Trejo at the premiere of Grindhouse in Austin, Texas.
Born	Dan Trejo (1944-05-16) May 16, 1944 (age 68) Los Angeles, California
Occupation	Actor
Years active	1984–present
Spouse	Debbie Shreve

Dan "Danny" Trejo (Spanish pronunciation: [ˈtɾexo]; born May 16, 1944) is an American actor who has appeared in numerous Hollywood films, most notably in "tough guy" roles as a villain or an anti-hero.

Early life[link]

Born Dan Trejo in the Echo Park neighborhood of Los Angeles, California,^[1][2] Trejo is the son of Alice Rivera and Dan Trejo, a construction worker.^[3][4] He is of Mexican descent.^[5] He is a second cousin of filmmaker Robert Rodriguez, though the two were unaware that they were related until the filming of Desperado.^[6][7] As a youth, he roamed the streets around his home area, committing various petty crimes with his uncle, and eventually became addicted to heroin. He was arrested numerous times as an adolescent.^{[citation needed]}

While on the streets, Trejo developed talent as a boxer and considered taking it up as a profession. That ambition was dashed by a lengthy prison sentence. He served 11 years for drug and robbery charges in the Holmesburg Prison^{[citation needed]}. While in Holmesburg, he became the Pennsylvania state prison champion in both the lightweight and welterweight divisions. During this time Trejo became a member of a twelve-step program which he credits with his success in overcoming drug addiction. He also served time in Folsom State Prison. Trejo was released in 1972.^[2]

Career[link]

While speaking at a recovery meeting in Los Angeles, Trejo met a young man in the movie business. His newfound friend invited him to the set of Runaway Train where Trejo was offered a job as an extra in the film's prison scenes. Edward Bunker, himself a former convict and well-respected crime author who was writing the screenplay for the film, recognized Trejo, with whom he had done time at San Quentin. It was widely misreported that Trejo was actually serving time in the prison where filming took place. Bunker remembered Trejo's boxing skills, and offered him $320 per day to train Eric Roberts, one of the movie's stars, for a boxing scene. Director Andrei Konchalovsky liked Trejo's work and decided to offer him a prominent role in the film.

Since then, Trejo has become a prolific actor in both films and television, acting alongside some of Hollywood's most famous actors, including Jean-Claude Van Damme, Johnny Depp, Al Pacino, Nicolas Cage, Charles Bronson, George Clooney, John Malkovich, Robert De Niro, Harrison Ford, Val Kilmer, and Antonio Banderas. Trejo has often appeared in five or more movies per year in various genres. Major releases in which he has acted include Machete, Blood in Blood Out, Animal Factory, Bubble Boy, Anaconda, Marked for Death, xXx, Desperado, Once Upon a Time in Mexico, Heat, From Dusk till Dawn, Con Air, Six Days Seven Nights, Spy Kids, The Replacement Killers, Anchorman: The Legend of Ron Burgundy, Predators, Urban Justice, The Devil's Rejects, Valley of Angels, Delta Farce, Grindhouse, Rob Zombie's Halloween and Snoop Dogg's Hood of Horror.

Trejo at Muscle Beach

Trejo is known for his very distinctive appearance. In addition to his heavily lined face (usually sporting a long mustache), he has displayed the large tattoo on his chest for many roles (the tattoo depicts a woman wearing a sombrero). He often plays thugs or lowlife characters, although he has also played protagonists in films such as Robert Rodriguez's Spy Kids trilogy, Bubble Boy and a guest appearance on the USA Network series Monk with art imitating life as a convict roommate to Tony Shalhoub's undercover detective. He also appeared in an episode of Desperate Housewives in Season 2, sent by Carlos to help Gabrielle deal with her grief of losing her unborn baby (another good-guy character). He voices the character Enrique, and Octavio's character is based on his appearance on King of the Hill. His role on King of the Hill was referenced in the HBO series Unscripted in which he appears in a quick cameo. He also is featured in an episode of Tim and Eric Awesome Show, Great Job! and an episode of Do IT LIVE alongside actors Anna Bennett and Claudia Pickering, shot in Downtown Los Angeles, by the name of Sebring feat. Danny Trejo. Trejo also made guest appearances in the award-winning series Breaking Bad in 2009 and 2010.

The award-winning independent film, CHAMPION was released in 2005, documenting Trejo's life. It features Dennis Hopper, Val Kilmer, Steve Buscemi and Robert Rodriguez. Trejo appears as a playable character in Def Jam: Fight for NY, a fighting game to which the actor lent his voice and likeness. He had previously lent his voice to Grand Theft Auto: Vice City and Grand Theft Auto: Vice City Stories for the character Umberto Robina, who also resembles Trejo. Trejo voiced the character of Rico in Rob Zombie's animated film The Haunted World of El Superbeasto as well as starring alongside Steven Seagal in Urban Justice. He is now portraying the Mexican bartender in the daytime soap opera The Young and the Restless. He also voices Raul the Ghoul in Fallout: New Vegas.

Regarding his continued growth as a professional actor, Trejo has stated that he still cannot believe how fortunate he has been. "I'm so blessed. I'm still scared that somebody's going to wake me up and say, 'Hey, we're still in prison. Let's go to chow'".^[8]

Many of Trejo's characters in Robert Rodriguez movies have been named after knives or sharp instruments: Machete in Spy Kids, Razor Eddie in the From Dusk till Dawn series, Navajas (Spanish for blades) in Desperado, and Cuchillo (Spanish for knife) in Predators.^[9] He also plays Machete in a trailer made for Rodriguez's film collaboration with Quentin Tarantino, Grindhouse. In 2010, he starred in a full theatrical version of Machete, based on the character.

He has also made cameo appearances in music videos such as, in rapper Tech N9ne's video for the single, Like Yeah, from his 2008 release Killer, Rehab's 2007 song Bartender, Jay Chou's short movie-music video Double Blade from his 2003 album Ye Hui Mei, in southern metalcore band Maylene and the Sons of Disaster's video for the single, Open Your Eyes, from their 2011 release IV and the Jonas Brothers' Burnin' Up from their album A Little Bit Longer and in Sepultura video Attitude in 1996.

He also appeared in the PlayStation Move game The Fight: Lights Out as an instructor for the player's character, and as himself in the second map pack for Call of Duty: Black Ops; Escalation map pack, on the zombie map Call of the Dead.

In 2012, Trejo starred alongside Ron Perlman and Charles S. Dutton in Craig Moss’s action film Bad Ass. He plays the main character of Frank Vega, a Vietnam veteran based upon 67-year-old “Epic Beard Man” Thomas Bruso, who became famous for successfully defending himself when attacked by two thugs who were bothering a 50-year old man aboard an AC Transit bus in Oakland, California.

Personal life[link]

Trejo is married to Debbie Shreve. He has four children: Danny Boy (b. 1981), Gilbert (b. 1988), Danielle (b. 1990), Jose (b. 1991) who are all from previous relationships. He also has a daughter Esmeralda Trejo who is married and has three children in Austin, Texas. Her mother is the woman with the sombrero on his chest tattoo.

Trejo has said that he is amazed when he walks the streets where he used to commit crimes and children ask him for autographs. He also pointed out that he often chooses to play bad guys to teach younger audiences that bad guys often die or go to prison, and that one should try to live a decent life.

Trejo often speaks in front of youth groups and at colleges and encourages young people to avoid making the same mistakes he did while growing up.

On February 7, 2012 Trejo visited John H. Francis Polytechnic high school to talk to students about education.

Trejo is a Christian.^[10]

Filmography[link]

Cultural references[link]

Plastilina Mosh, a Mexican alternative rock band, paid tribute to Trejo with their song "Danny Trejo", featured in their album All U Need Is Mosh. Trejo is also featured in the 2004 hip-hop fighting game Def Jam: Fight for NY as a soldier to Crow's side. Crow is voiced by and his looks are captured by Snoop Dogg. Danny Trejo can also be seen in the intro of Greg Hastings Tournament Paintball Max'd ("Play for Real", B-Real & DJ Lethal). He also is featured in the downloadable content pack called Escalation Map Pack in the video game Call of Duty: Black Ops on the map "Call of the Dead" as himself.

ITN Clothing[link]

In addition to his acting career, Trejo is also the founder of ITN Clothing, which offers hip-hop style T-shirts and accessories.

References[link]

External links[link]

Wikimedia Commons has media related to: Danny Trejo

• Danny Trejo at the Internet Movie Database
• Danny Trejo at TV Tropes
• Danny Trejo: The illustrated man
• Official site for Champion a documentary about Trejo
• Random Roles: Danny Trejo
• Danny Trejo with Estevan Oriol