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Revisi per 19 Desember 2014 18.06

Permukaan (Inggris: surface) dalam matematika, khususnya dalam topologi, adalah suatu kelipatan topologi dalam dua dimensi. Contoh paling umum adalah batas suatu benda padat dalam ruang tiga dimensi biasa R³ — misalnya, permukaan suatu bola. Di sisi lain, ada permukaan-permukaan, seperti "botol Klein", yang tidak dapat dimasukkan ke dalam ruang Euklidean tiga dimensi tanpa menyertakan singularitas atau potongan ke diri sendiri.

Definisi

Suatu permukaan (secara topologi) adalah suatu ruang topologi yang setiap titiknya mempunyai satu tetangga homeomorfik terbuka terhadap sejumlah subset terbuka pada bidang Euklidean E². Tetangga semacam itu, bersama dengan homeomorfisme terkait, dikenal sebagai "peta (koordinat)" (coordinate chart). Melalui peta ini maka tetangga itu mendapatkan koordinat standar pada bidang Euklidean. Koordinat-koordinat ini dikenal sebagai koordinat lokal dan homeomorfisme ini membuat permukaan itu dikatakan sebagai secara lokal Euklidean.

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 bidang proyeksi real adalah contoh-contoh dari permukaan tertutup.

Lihat pula

Luas permukaan
Bentuk volume, untuk volume permukaan dalam Eⁿ
Poincaré metric, for metric properties of Riemann surfaces
Area element, the area of a differential element of a surface
Roman surface
Boy's surface
Tetrahemiheksahedron

Catatan

Referensi

Dyck, Walther (1888), "Beiträge zur Analysis situs I", Math. Ann., 32: 459–512, doi:10.1007/bf01443580
Gramain, André (1984). Topology of Surfaces. BCS Associates. ISBN 0-914351-01-X. (Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")

A. Champanerkar; et al., Classification of surfaces via Morse Theory (PDF), an exposition of Gramain's notes
Bredon, Glen E. (1993). Topology and Geometry. Springer-Verlag. ISBN 0-387-97926-3.
Massey, William S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag. ISBN 0-387-97430-X.
Francis, George K.; Weeks, Jeffrey R. (May 1999), "Conway's ZIP Proof" (PDF), American Mathematical Monthly, 106 (5), page discussing the paper: On Conway's ZIP Proof

Pranala luar

The Classification of Surfaces and the Jordan Curve Theorem in Home page of Andrew Ranicki
Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing
Math Surfaces Animation, with JavaScript (Canvas HTML) for tens surfaces rotation viewing
The Classification of Surfaces Lecture Notes by Z.Fiedorowicz
History and Art of Surfaces and their Mathematical Models
2-manifolds at the Manifold Atlas

@@ Baris 7: / Baris 7: @@
 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==
-A ''(topological) surface'' is a [[topological space]] in which every point has an open [[topological neighbourhood|neighbourhood]] [[homeomorphism|homeomorphic]] to some [[open set|open subset]] of the Euclidean plane '''E'''<sup>2</sup>. Such a neighborhood, together with the corresponding homeomorphism, is known as a ''(coordinate) chart''. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as ''local coordinates'' and these homeomorphisms lead us to describe surfaces as being ''locally Euclidean''.
+Suatu ''permukaan'' (secara topologi) adalah suatu ruang topologi yang setiap titiknya mempunyai satu [[:en:topological neighbourhood|tetangga]] [[:en:homeomorphism|homeomorfik]] terbuka terhadap sejumlah [[:en:open set|subset terbuka]] pada bidang Euklidean '''E'''<sup>2</sup>. Tetangga semacam itu, bersama dengan homeomorfisme terkait, dikenal sebagai "peta (koordinat)" (''coordinate chart''). Melalui peta ini maka tetangga itu mendapatkan koordinat standar pada bidang Euklidean. Koordinat-koordinat ini dikenal sebagai ''koordinat lokal'' dan homeomorfisme ini membuat permukaan itu dikatakan sebagai ''secara lokal Euklidean''.
+<!--
 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.
@@ Baris 16: / Baris 16: @@
 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).