Permukaan: Perbedaan revisi

93 bita ditambahkan ,  5 tahun yang lalu
tidak ada ringkasan suntingan
(←Membuat halaman berisi '{{About|permukaan dari sudut pandang topologi}} thumb|225px|right|Suatu "permukaan terbuka" dengan kontour ''X''-, ''Y''-, dan ''Z''. '''Permu...')
 
 
The concept of surface finds application in [[physics]], [[engineering]], [[computer graphics]], and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the [[aerodynamics|aerodynamic]] properties of an [[airplane]], the central consideration is the flow of air along its surface.
-->
 
==Definisi ==
==Definitions and first examples==
ASuatu ''(topological) surfacepermukaan'' is(secara atopologi) [[topologicaladalah space]]suatu inruang whichtopologi everyyang pointsetiap hastitiknya anmempunyai opensatu [[:en:topological neighbourhood|neighbourhoodtetangga]] [[:en:homeomorphism|homeomorphichomeomorfik]] toterbuka someterhadap sejumlah [[:en:open set|open subset terbuka]] ofpada thebidang Euclidean planeEuklidean '''E'''<sup>2</sup>. SuchTetangga asemacam neighborhooditu, togetherbersama withdengan thehomeomorfisme corresponding homeomorphismterkait, isdikenal knownsebagai as"peta a(koordinat)" (''(coordinate) chart''). ItMelalui ispeta throughini thismaka charttetangga thatitu themendapatkan neighborhoodkoordinat inheritsstandar thepada standardbidang coordinates on the Euclidean planeEuklidean. These coordinatesKoordinat-koordinat areini knowndikenal assebagai ''localkoordinat coordinateslokal'' anddan thesehomeomorfisme homeomorphismsini leadmembuat uspermukaan toitu describedikatakan surfaces as beingsebagai ''locallysecara lokal EuclideanEuklidean''.
<!--
 
In most writings on the subject, it is often assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, [[Second-countable space|second countable]], and [[Hausdorff space|Hausdorff]]. It is also often assumed that the surfaces under consideration are connected.
 
 
More generally, a ''(topological) surface with boundary'' is a [[Hausdorff space|Hausdorff]] [[topological space]] in which every point has an open [[topological neighbourhood|neighbourhood]] [[homeomorphism|homeomorphic]] to some [[open set|open subset]] of the closure of the [[upper half-plane]] '''H'''<sup>2</sup> in '''C'''. These homeomorphisms are also known as ''(coordinate) charts''. The boundary of the upper half-plane is the ''x''-axis. A point on the surface mapped via a chart to the ''x''-axis is termed a ''boundary point''. The collection of such points is known as the ''boundary'' of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the ''x''-axis is an ''interior point''. The collection of interior points is the ''interior'' of the surface which is always non-[[empty set|empty]]. The closed [[disk (mathematics)|disk]] is a simple example of a surface with boundary. The boundary of the disc is a circle.
-->
 
Istilah "permukaan" tanpa tambahan kualifikasi merujuk kepada permukaan tanpa batasan. Terutama, suatu permukaan dengan batasan kosong adalah permukaan dalam arti umum. Suatu permukaan dengan batasan kosong yang kompak dikenal sebagai "permukaan tertutup" (''closed surface''). Bola dua dimensi, [[torus]] dua dimensi, dan [[:en:real projective plane|bidang proyeksi real]] adalah contoh-contoh dari permukaan tertutup.
The term ''surface'' used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional [[torus]], and the [[real projective plane]] are examples of closed surfaces.
<!--
 
The [[Möbius strip]] is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be ''orientable'' if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not (because the real projective plane with one point removed is homeomorphic to the open Möbius strip).