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{{disambiginfo|Titik (disambiguasi)}}Dalam [[geometri Euklides]], '''titik''' adalah suatu gagasan primitif yang memodelkan lokasi yang tepat di dalam [[Ruang Euklides|ruang]], serta tidak memiliki panjang, lebar, atau kedalaman.{{sfnp|Ohmer|1969|p=34–37}} Gagasan primitif pada konteks ini berarti bahwa suatu titik tidak dapat didefinisikan dalam objek yang didefinisikan sebelumnya, dalam artian bahwa titik hanya didefinisikan dengan beberapa [[aksioma]] yang harus terpenuhi. Titik dalam matematika yang modern lebih mengacu pada suatu [[Anggota (matematika)|anggota]] dari suatu [[Himpunan (matematika)|himpunan]] yang dikenal dengan sebutan [[Ruang (matematika)|ruang]].
Di dalam [[geometri]], [[topologi]], dan cabang-cabang matematika yang saling berkaitan, sebuah '''titik spasial''' menggambarkan objek yang spesifik di dalam ruang yang diberikan, yang tidak melibatkan [[volume]], [[luas]], [[panjang]], atau analog-analog lainnya pada [[dimensi]] yang lebih tinggi. Dengan demikian, titik adalah objek 0-dimensi. Karena sifatnya sebagai salah satu konsep geometri paling sederhana, ia sering digunakan di dalam satu bentuk atau bentuk lain sebagai konstituen dasar geometri, [[fisika]], [[grafik vektor]], dan banyak lapangan lainnya.
 
==''' Titik di dalam geometri Euclidean'''Euklides ==
[[Berkas:ACP_3.svg|jmpl|Suatu himpunan berhingga dari titik-titk (biru) di dalam [[ruang Euklides]] dua dimensi.]]
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[[Image:ACP_3.svg|thumb|A finite set of points (blue) in two dimensional [[Euclidean space]].]]
 
PointsTitik, areyang mostsering oftendipandang considereddi withindalam thekerangka framework ofkerja [[Euclideangeometri geometryEuklides]], wheremerupakan theysalah aresatu oneobjek ofyang thepaling fundamental objectsmendasar. [[Euclid|Euklides]] originallymulanya definedmendefinisikan thetitik point vaguely, assebagai "thatobjek whichyang hastidak nomemiliki partbagian".{{sfnp|Heath|1956|p=153}} In two dimensionalDalam [[Euclideanruang spaceEuklides]], adua point isdimensi, representedtitik bydinyatakan ansebagai [[orderedpasangan pairterurut]], <math>\, (x,y)</math>,; ofbilangan numbers,pertama wherepada thepasangan firsttersebut, numbermenurut [[Conventionkonvensi, (norm)|conventionally]]menyatakan represents the [[horizontal]] anddan issering oftendituliskan denoted bysebagai <math>\, x</math>, andsementara thebilangan secondkedua number conventionally represents the [[vertical]]menyatakan andvertikal isdan oftensering denoteddituliskan bysebagai <math>\, y</math>. Gagasan Thisini ideamudah isdiperumum easilyke generalizeddalam toruang threeEuklides dimensionaltiga Euclidean spacedimensi, wheredengan atitik pointdinyatakan isoleh representedpasangan byterurut an ordered triplet,rangkap tiga<math>\, (x,y,z)</math>, withdengan thebilangan additionaltambahan thirdketiga numbermenyatakan representingkedalaman depthdan anddinyatakan often denoted bydengan <math>z</math>. Perumuman Furtherlebih generalizationslanjut aredinyatakan representeddengan bypasangan anterurut ordered tuplet ofrangkap <math>n terms</math>,<math>\, (a_1,a_2,...,a_n)</math>, wheredengan <math>n</math> isadalah thedimensi dimensionruang oftempat thetitik space in which the point is locatedberada.{{sfnp|Silverman|1969|p=7}}
 
ManyBanyak constructsobjek withinyang Euclideandibangun geometrydi consistdalam ofgeometri anEuklides terdiri dari [[infinitytak hingga|infinitetak berhingga]] collectionbanyaknya ofkumpulan pointstitik-titik thatyang conformsesuai todengan certainaksioma-aksioma axiomstertentu. Hal This is usuallyini representedbiasanya bydinyatakan aoleh [[Sethimpunan (mathematicsmatematika)|sethimpunan]] of pointstitik-titik; As an examplemisalnya, a [[linegaris (mathematicsgeometri)|linegaris]] isadalah anhimpunan infinitetak sethingga ofbanyaknya pointstitik-titik of the formyang berbentuk<math display="block">\, L = \lbrace (a_1,a_2,...a_n)|a_1c_1 + a_2c_2 + ... a_nc_n = d \rbrace, </math>, wheredengan <math>\, c_1</math> throughmelalui <math>\, c_n</math> anddan <math>\, d</math> areadalah constantskonstanta, andserta <math>n</math> is the dimension ofadalah thedimensi spaceruang. Juga Similarterdapat constructionskonstruksi-konstruksi existserupa thatyang define themendefinisikan [[planebidang (mathematicsgeometri)|planebidang]], [[lineruas segmentgaris]], anddan otherkonsep-konsep relatedlainnya conceptsyang saling berkaitan.{{sfnp|de Laguna|1922}}
 
== Geometri tanpa titik ==
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line. This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's postulation of points was neither complete nor definitive, as he occasionally assumed facts about points that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.
Titik sudah dianggap merupakan gagasan yang fundamental dalam [[geometri]] dan [[topologi]]. Meskipun demikian, terdapat beberapa cabang yang tidak menggunakan gagasan titik, seperti [[geometri nonkomutatif]] (''noncommutative geometry'') dan [[topologi bebas titik]] (''pointless topology''). “Ruang bebas titik” (''pointfree space'') atau "ruang tanpa titik" (''pointless space'') tidak didefinisikan sebagai [[himpunan (matematika)|himpunan]], melainkan didefinisikan melalui beberapa struktur ([[C*-aljabar|aljabar]] atau [[aljabar Heyting lengkap|logika]]) yang terlihat seperti ruang fungsi yang terkenal pada himpunan tersebut, yaitu aljabar dari [[fungsi kontinu]] atau [[aljabar himpunan]]. Lebih tepatnya, struktur tersebut memperumum ruang yang terkenal dari [[fungsi]] menurut suatu cara di mana operasi “mengambil nilai pada titik tersebut” dapat didefinisikan.{{sfnp|Gerla|1985}}
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== Lihat pula ==
== Titik di dalam cabang-cabang matematika ==
* [[Titik limit]]
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* [[Ruang afin]]
A point in [[point-set topology]] is defined as a member of the underlying set of a [[topological space]].
* [[Batas (topologi)|Titik batas]]
* [[Titik kritis (matematika)|Titik kritis]]
* [[Titik puncak (singularitas)|Titik puncak]]
* [[Singularitas|Titik singular]]
 
== Catatan ==
Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forego it, e.g. [[noncommutative geometry]] and [[pointless topology]]. A “pointless space” is defined not as a [[set (mathematics)|set]], but via some structure ([[C*-algebra|algebraical]] or [[complete Heyting algebra|logical]] respectively) which looks like a well-known function space on the set: an algebra of [[continuous function]]s or an [[algebra of sets]] respectively. More precisely, such structures generalize well-known spaces of [[function]]s in a way that the operation “take a value at this point” may not be defined.
{{Reflist|colwidth=30em}}
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== Lihat pulaReferensi ==
{{refbegin}}
<!--
{{div col|colwidth=30em}}
* [[Accumulation point]]
*{{cite journal
* [[Affine space]]
|last = de Laguna |first = T. |author-link = Theodore de Laguna
* [[Boundary point]]
|year = 1922
* [[critical point (mathematics)|Critical point]]
|title = Point, line and surface as sets of solids,
* [[Cusp (singularity)|Cusp]]
|journal = The Journal of Philosophy
* [[Singular point]]
|volume = 19
-->
|issue = 17
|pages = 449–461.
|jstor = 2939504
|doi = 10.2307/2939504}}
*{{cite book
|last = Gerla
|first = G
|year = 1995
|contribution-url = http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf
|contribution = Pointless Geometries
|editor1-last = Buekenhout
|editor1-first = F.
|editor2-last = Kantor
|editor2-first = W
|title = Handbook of Incidence Geometry: Buildings and Foundations
|publisher = North-Holland
|page = 1015–1031.
|access-date = 2023-02-26
|archive-date = 2011-07-17
|archive-url = https://web.archive.org/web/20110717210751/http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf
|dead-url = yes
}}
*{{cite book
|last = Heath |first = Thomas L.
|author-link = Thomas Little Heath
|title = The Thirteen Books of Euclid's Elements
|volume = 1
|edition = 2nd
|year = 1956
|publisher = Dover Publications
|location = New York
|isbn = 0-486-60088-2
|url = https://archive.org/details/thirteenbooksofe00eucl
}}
*{{cite book
|last = Silverman |first = Richard A.
|title = Modern Calculus and Analytic Geometry
|url = https://books.google.com/books?id=DcWHAwAAQBAJ&pg=PA7
|year = 1969
|publisher = Macmillan}}
{{div col end}}
{{refend}}
 
== Pranala luar ==
* [http://www.mathopenref.com/point.html Definisi Titik] dengan applet interaktif
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* [http://www.mathopenref.com/pointtocs/pointstoc.html DefinitionHalaman ofdefinisi Pointtitik], withdengan interactiveanimasi appletinteraktif yang juga berguna di dalam suasana ruang kelas. Math Open Reference
{{bangun}}
* [http://www.mathopenref.com/tocs/pointstoc.html Points definition pages], with interactive animations that are also useful in a classroom setting. Math Open Reference
 
* {{planetmath reference|id=8173|title=Point}}
{{Authority control}}
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[[Kategori:Geometri]]
[[Kategori:Matematika]]
 
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