open set

In a metric space M a set O is called an open subset of M or just open, if for every xO there is an open ball S around x such that SO. If d(x,y) is the distanceMathworldPlanetmath from x to y then the open ball Br with radius r>0 around x is given as:


Using the idea of an open ball one can define a neighborhoodMathworldPlanetmathPlanetmath of a point x. A set containing x is called a neighborhood of x if there is an open ball around x which is a subset of the neighborhood.

These neighborhoods have some properties, which can be used to define a topological spaceMathworldPlanetmath using the Hausdorff axioms for neighborhoods, by which again an open set within a topological space can be defined. In this way we drop the metric and get the more general topological space. We can define a topological space X with a set of neighborhoods of x called Ux for every xX, which satisfy

  1. 1.

    xU for every UUx

  2. 2.

    If UUx and VX and UV then VUx (every set containing a neighborhood of x is a neighborhood of x itself).

  3. 3.

    If U,VUx then UVUx.

  4. 4.

    For every UUx there is a VUx, such that VU and VUp for every pV.

The last point leads us back to open sets, indeed a set O is called open if it is a neighborhood of every of its points. Using the properties of these open sets we arrive at the usual definition of a topological space using open sets, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the above definition. In this definition we look at a set X and a set of subsets of X, which we call open sets, called 𝒪, having the following properties:

  1. 1.

    𝒪 and X𝒪.

  2. 2.

    Any union of open sets is open.

  3. 3.

    intersectionsMathworldPlanetmath of open sets are open.

Note that a topological space is more general than a metric space, i.e. on every metric space a topology can be defined using the open sets from the metric, yet we cannot always define a metric on a topological space such that all open sets remain open.


  • On the real axis the intervalMathworldPlanetmathPlanetmath I=(0,1) is open because for every aI the open ball with radius min(a,1-a) is always a subset of I. (Using the standard metric d(x,y)=|x-y|.)

  • The open ball Br around x is open. Indeed, for every yBr the open ball with radius r-d(x,y) around y is a subset of Br, because for every z within this ball we have:


    So d(x,z)<r and thus z is in Br. This holds for every z in the ball around y and therefore it is a subset of Br

  • A non-metric topology would be the finite complementMathworldPlanetmath topology on infinite setsMathworldPlanetmath, in which a set is called open, if its complement is finite.

Title open set
Canonical name OpenSet
Date of creation 2013-03-22 12:39:25
Last modified on 2013-03-22 12:39:25
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 21
Author mathwizard (128)
Entry type Definition
Classification msc 54A05
Synonym open
Synonym open subset
Defines Hausdorff axioms