topology

English

Etymology

From Late Latin topologia, from Ancient Greek τόπος (tópos, “place, locality”) + -(o)logy (“study of, a branch of knowledge”).

Pronunciation

• (Received Pronunciation) IPA^(key): /təˈpɒləd͡ʒi/
• (General American) IPA^(key): /təˈpɑləd͡ʒi/

  • Audio (US):

    (file)

• Rhymes: -ɒlədʒi

Noun

topology (countable and uncountable, plural topologies)

1. (mathematics, uncountable) The branch of mathematics dealing with those properties of a geometrical object (of arbitrary dimensionality) that are unchanged by continuous deformations (such as stretching, bending, etc., without tearing or gluing).
   • 1970 [Addison-Wesley], Stephen Willard, General Topology, 2012, Dover, page v,

     This book is designed to develop the fundamental concepts of general topology which are the basic tools of working mathematicians in a variety of fields.

   • 1974 [Crane, Russak & Co.], H. Graham Flegg, From Geometry to Topology, 2001, Dover, page v,

     Many university courses in topology plunge immediately into a formalized and entirely abstract presentation of topological concepts.

2. (topology) Any collection τ of subsets of a given set X that contains both the empty set and X, and which is closed under finitary intersections and arbitrary unions.

   A set $X$ equipped with a topology $\tau$ is called a topological space and denoted $(X,\tau )$.

   The subsets of a set $X$ which constitute a topology are called the open sets of $X$.

   • 2016, Ehud Hrushovski, François Loeser, Non-Archimedean Tame Topology and Stably Dominated Types (AM-192), Princeton University Press, page 43:

     It is easy to verify that the topology generated by these basic open sets coincides with the definition of the topology on ${\widehat {V}}$ above, for the Zariski topology and the sheaf of functions $\operatorname {val} (f),f$ regular.

3. (medicine) The anatomical structure of part of the body.
4. (computing) The arrangement of nodes in a communications network.
5. (technology) The properties of a particular technological embodiment that are not affected by differences in the physical layout or form of its application.
6. (topography) The topographical study of geographic locations or given places in relation to their history.
7. (dated) The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place.

Synonyms

• (branch of mathematics): analysis situs (obsolete), geometria situs (obsolete)

Hypernyms

• (collection of subsets): pi-system, π-system

Meronyms

• open set

Holonyms

• topological space

Derived terms

• Alexandrov topology
• algebraic topology
• bus topology
• cofinite topology
• combinatorial topology
• differential topology
• discrete topology
• disjoint union topology
• ditopology
• electrotopology
• final topology
• geometric topology
• Grothendieck topology
• heterotopology
• hypertopology
• initial topology
• link topology
• mereotopology
• microtopology
• nanotopology
• network topology
• pointless topology
• point-set topology
• pretopology
• product topology
• psychotopology
• relative topology
• retopology
• ring topology
• Scott topology
• standard topology
• star topology
• subspace topology
• subtopology
• trivial topology
• Zariski topology

 

Related terms

• topological
• topologically
• topological space
• topologist

Translations

study of geometric properties that are not changed by stretching etc.

• Afrikaans: topologie
• Catalan: topologia (ca) f
• Chinese:

  Mandarin: 拓撲學／拓扑学 (zh) (tuòpūxué)

• Czech: topologie (cs) f
• Danish: topologi (da)
• Dutch: topologie (nl) f
• Esperanto: topologio
• Estonian: topoloogia
• Finnish: topologia (fi)
• French: topologie (fr) f
• Galician: topoloxía f
• German: Topologie (de) f
• Greek: τοπολογία (el) f (topología)
• Hebrew: טופולוגיה (he) f (topológya)
• Hungarian: topológia (hu)
• Icelandic: grannfræði (is) f, stöðfræði f
• Irish: toipeolaíocht f
• Italian: topologia (it) f
• Japanese: 位相幾何学 (いそうきかがく, isō-kikagaku), 位相数学 (いそうすうがく, isō-sūgaku), トポロジー (toporojī)
• Kazakh: топология (topologiä)
• Khmer: ឋានវិទ្យា (thaanvɨtyiə)
• Korean: 위상수학 (ko) (wisangsuhak)
• Latvian: topoloģija f
• Macedonian: топологија f (topologija)
• Polish: topologia (pl) f
• Portuguese: topologia (pt) f
• Romanian: topologie (ro) f
• Russian: тополо́гия (ru) f (topológija)
• Serbo-Croatian: topologija (sh) f, топологија f
• Sicilian: topoluggìa f
• Slovene: topologija f
• Spanish: topología f
• Ukrainian: тополо́гія (uk) f (topolóhija)
• Vietnamese: tô pô học

anatomical structure

• Greek: τοπολογία (el) f (topología)
• Russian: тополо́гия (ru) f (topológija)
• Serbo-Croatian: topologija (sh) f, топологија f

arrangements of computer nodes

• Chinese:

  Mandarin: 拓撲／拓扑 (zh) (tuòpū)

• Czech: topologie (cs) f
• Finnish: topologia (fi)
• French: topologie (fr) f
• German: Topologie (de) f
• Greek: τοπολογία (el) f (topología)
• Hungarian: topológia (hu)
• Japanese: トポロジー (toporojī)
• Maori: āhuawhatunga
• Portuguese: topologia (pt) f
• Russian: тополо́гия (ru) f (topológija)
• Serbo-Croatian: topologija (sh) f, топологија f

See also

• homotopy
• Appendix:Glossary of topology

Further reading

• topology on Wikipedia.Wikipedia
• Homotopy on Wikipedia.Wikipedia

Anagrams

• optology