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{{for|the concept in graph theory|Neighbourhood (graph theory)}}
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you adored this post and you would certainly such as to receive even more information regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] kindly browse through our web site.
[[File:Neighborhood illust1.png|right|thumb|A set <math>V</math> in the [[plane (geometry)|plane]] is a neighbourhood of a point <math>p</math> if a small disk around <math>p</math> is contained in <math>V</math>.]]
[[File:Neighborhood illust2.svg|right|thumb|A rectangle is not a neighbourhood of any of its corners.]]
 
In [[topology]] and related areas of [[mathematics]], a '''neighbourhood''' (or '''neighborhood''') is one of the basic concepts in a [[topological space]]. Intuitively speaking, a neighbourhood of a point is a [[Set (mathematics)|set]] containing the point where you can move that point some amount without leaving the set.
 
This concept is closely related to the concepts of [[open set]] and [[interior (topology)|interior]].
 
== Definition ==
 
If <math>X</math> is a [[topological space]] and <math>p</math> is a point in <math>X</math>, a '''neighbourhood''' of <math>p</math> is a [[subset]] <math>V</math> of <math>X</math>, which includes an [[open set]] <math>U</math> containing <math>p</math>,
:<math>p \in U \subseteq V.</math>
 
This is also equivalent to <math>p\in X</math> being in the [[Interior (topology)#Interior_point|interior]] of <math>V</math>.
 
Note that the neighbourhood <math>V</math> need not be an open set itself. If <math>V</math> is open it is called an '''open neighbourhood'''. Some [[scholar]]s require that neighbourhoods be open, so it is important to note conventions.
 
A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.
 
The collection of all neighbourhoods of a point is called the [[neighbourhood system]] at the point.
 
If <math>S</math> is a [[subset]] of <math>X</math> then a '''neighbourhood''' of <math>S</math> is a set <math>V</math> which includes an open set <math>U</math> containing <math>S</math>. It follows that a set <math>V</math> is a neighbourhood of <math>S</math> if and only if it is a neighbourhood of all the points in <math>S</math>. Furthermore, it follows that <math>V</math> is a neighbourhood of <math>S</math> [[iff]] <math>S</math> is a subset of the [[Interior (topology)|interior]] of <math>V</math>.
 
== In a metric space ==
[[File:Neighborhood illust3.png|right|thumb|A set <math>S</math> in the plane and a uniform neighbourhood <math>V</math> of <math>S</math>.]]
In a [[metric space]] <math>M = (X, d)</math>, a set <math>V</math> is a '''neighbourhood''' of a point <math>p</math> if there exists an [[open ball]] with centre <math>p</math> and radius <math>r>0</math>, such that
:<math>B_r(p) = B(p;r) = \{ x \in X \mid d(x,p) < r \}</math>
is contained in <math>V</math>.
 
<math>V</math> is called '''uniform neighbourhood''' of a set <math>S</math> if there exists a positive number <math>r</math> such that for all elements <math>p</math> of <math>S</math>,
:<math>B_r(p) = \{ x \in X \mid d(x,p) < r \}</math>
is contained in <math>V</math>.
 
For <math>r > 0</math> the '''<math>r</math>-neighbourhood''' <math>S_r</math> of a set <math>S</math> is the set of all points in <math>X</math> which are at distance less than <math>r</math> from <math>S</math> (or equivalently, <math>S</math><sub><math>r</math></sub> is the union of all the open balls of radius <math>r</math> which are centred at a point in <math>S</math>).
 
It directly follows that an <math>r</math>-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an <math>r</math>-neighbourhood for some value of <math>r</math>.
 
== Examples ==
 
Given the set of [[real number]]s <math>\mathbb{R}</math> with the usual [[Euclidean metric]] and a subset <math>V</math> defined as
:<math>V:=\bigcup_{n \in \mathbb{N}} B\left(n\,;\,1/n \right),</math>
then <math>V</math> is a neighbourhood for the set <math>\mathbb{N}</math> of [[natural number]]s, but is ''not'' a uniform neighbourhood of this set.
 
—
[[Epsilon-neighborhood]]
== Topology from neighbourhoods ==
 
The above definition is useful if the notion of [[open set]] is already defined. There is an alternative way to define a topology, by first defining the [[neighbourhood system]], and then open sets as those sets containing a neighbourhood of each of their points.
 
A neighbourhood system on <math>X</math> is the assignment of a [[filter (mathematics)|filter]] <math>N(x)</math> (on the set <math>X</math>) to each <math>x</math> in <math>X</math>, such that
# the point <math>x</math> is an element of each <math>U</math> in <math>N(x)</math>
# each <math>U</math> in <math>N(x)</math> contains some <math>V</math> in <math>N(x)</math> such that for each <math>y</math> in <math>V</math>, <math>U</math> is in <math>N(y)</math>.
 
One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.
 
== Uniform neighbourhoods ==
 
In a [[uniform space]] <math>S = (X, \delta)</math>, <math>V</math> is called a '''uniform neighbourhood''' of <math>P</math> if <math>P</math> is not [[closeness (mathematics)|close]] to <math>X \setminus V</math>, that is there exists no [[entourage (topology)|entourage]] containing <math>P</math> and <math>X \setminus V</math>.
 
==Punctured neighbourhood==
 
A '''punctured neighbourhood''' of a point <math>p</math> (sometimes called a '''deleted neighbourhood''') is a neighbourhood of <math>p</math>, without <math>\{p\}</math>. For instance, the [[interval (mathematics)|interval]] <math>(-1, 1) = \{y : -1 < y < 1\}</math> is a neighbourhood of <math>p = 0</math> in the [[real line]], so the set <math>(-1, 0) \cup (0, 1) = (-1, 1) \setminus \{0\}</math> is a punctured neighbourhood of <math>0</math>. Note that a punctured neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of punctured neighbourhood occurs in the [[Limit_of_a_function#Functions_on_topological_spaces|definition of the limit of a function]].
 
==See also==
 
* [[Tubular neighborhood]]
 
==References==
*{{cite book
| last      = Kelley
| first      = John L.
| title      = General topology
| publisher  = New York: Springer-Verlag
| year      = 1975
| pages      =
| isbn      = 0-387-90125-6
}}
*{{cite book
| last      = Bredon
| first      = Glen E.
| authorlink = Glen Bredon
| title      = Topology and geometry
| publisher  = New York: Springer-Verlag
| year      = 1993
| pages      =
| isbn      = 0-387-97926-3
}}
*{{cite book
| last      = Kaplansky
| first      = Irving
| authorlink = Irving Kaplansky
| title      = Set Theory and Metric Spaces
| publisher  = American Mathematical Society
| year      = 2001
| pages      =
| isbn      = 0-8218-2694-8
}}
 
[[Category:General topology]]
[[Category:Mathematical analysis]]
 
[[pt:Vizinhança]]

Latest revision as of 09:51, 17 September 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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