Summary
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Let be a finite set. A topology on is a subset of (the power set of ) such that and . if then . if then . In other words, a subset of is a topology if contains both and and is closed under arbitrary unions and intersections. Elements of are called open sets. The general description of topological spaces requires that a topology be closed under arbitrary (finite or infinite) unions of open sets, but only under intersections of finitely many open sets. Here, that distinction is unnecessary. Since the power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets). A topology on a finite set can also be thought of as a sublattice of which includes both the bottom element and the top element . There is a unique topology on the empty set ∅. The only open set is the empty one. Indeed, this is the only subset of ∅. Likewise, there is a unique topology on a singleton set {a}. Here the open sets are ∅ and {a}. This topology is both discrete and trivial, although in some ways it is better to think of it as a discrete space since it shares more properties with the family of finite discrete spaces. For any topological space X there is a unique continuous function from ∅ to X, namely the empty function. There is also a unique continuous function from X to the singleton space {a}, namely the constant function to a. In the language of the empty space serves as an initial object in the while the singleton space serves as a terminal object. Let X = {a,b} be a set with 2 elements.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (3)
MATH-497: Homotopy theory
We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspen
MATH-410: Riemann surfaces
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
MATH-301: Ordinary differential equations
Le cours donne une introduction à la théorie des EDO, y compris existence de solutions locales/globales, comportement asymptotique, étude de la stabilité de points stationnaires et applications, en pa
Related publications (20)

Crossover between distinct symmetries in solid solutions of rare earth nickelates

Duncan Alexander, Bernat Mundet, Jean-Marc Triscone

A strong coupling of the lattice to functional properties is observed in many transition metal oxide systems, such as the ABO(3) perovskites. In the quest for tailor-made materials, it is essential to be able to control the structural properties of the com ...
AIP Publishing2021

Evolution of Topics and Novelty in Science

Orion B Penner

Methods of estimating the similarity between individual publications is an area of long-standing interest in the scientometrics community. Traditional methods have generally relied on references and other metadata, while text mining approaches based on tit ...
INT SOC SCIENTOMETRICS & INFORMETRICS-ISSI2019
Show more
Related concepts (16)
Alexandrov topology
In topology, an Alexandrov topology is a topology in which the intersection of every family of open sets is open. It is an axiom of topology that the intersection of every finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov topologies are uniquely determined by their specialization preorders.
Excluded point topology
In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named: If X has two points, it is called the Sierpiński space. This case is somewhat special and is handled separately.
List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. Discrete topology − All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.
Show more