open set (original) (raw)

In a metric space M a set O is called an open subset of M or just open, if for every x∈O there is an open ball S around x such that S⊂O. If d⁢(x,y) is the distance 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 neighborhood 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 space 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 x∈X, which satisfy

1. 1. x∈U for every U∈Ux
2. 1. If U∈Ux and V⊂X and U⊂V then V∈Ux (every set containing a neighborhood of x is a neighborhood of x itself).
3. 1. If U,V∈Ux then U∩V∈Ux.
4. 1. For every U∈Ux there is a V∈Ux, such that V⊂U and V∈Up for every p∈V.

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 equivalent 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 X∈𝒪.
2. 1. Any union of open sets is 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.

Examples:

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  On the real axis the interval I=(0,1) is open because for every a∈I 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 y∈Br 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:

  d⁢(x,z)≤d⁢(x,y)+d⁢(y,z)<d⁢(x,y)+r-d⁢(x,y)=r.
  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 complement topology on infinite sets, in which a set is called open, if its complement is finite.