open set
In a metric space $M$ a set $O$ is called an open subset of $M$ or just open, if for every $x\in O$ there is an open ball $S$ around $x$ such that $S\subset O$. If $d(x,y)$ is the distance^{} from $x$ to $y$ then the open ball ${B}_{r}$ 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 ${U}_{x}$ for every $x\in X$, which satisfy

1.
$x\in U$ for every $U\in {U}_{x}$

2.
If $U\in {U}_{x}$ and $V\subset X$ and $U\subset V$ then $V\in {U}_{x}$ (every set containing a neighborhood of $x$ is a neighborhood of $x$ itself).

3.
If $U,V\in {U}_{x}$ then $U\cap V\in {U}_{x}$.

4.
For every $U\in {U}_{x}$ there is a $V\in {U}_{x}$, such that $V\subset U$ and $V\in {U}_{p}$ for every $p\in V$.
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 $\mathcal{O}$, having the following properties:

1.
$\mathrm{\varnothing}\in \mathcal{O}$ and $X\in \mathcal{O}$.

2.
Any union of open sets is open.

3.
intersections^{} 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.
Examples:

•
On the real axis the interval^{} $I=(0,1)$ is open because for every $a\in I$ the open ball with radius $\mathrm{min}(a,1a)$ is always a subset of $I$. (Using the standard metric $d(x,y)=xy$.)

•
The open ball ${B}_{r}$ around $x$ is open. Indeed, for every $y\in {B}_{r}$ the open ball with radius $rd(x,y)$ around y is a subset of ${B}_{r}$, because for every $z$ within this ball we have:
$$ So $$ and thus $z$ is in ${B}_{r}$. This holds for every $z$ in the ball around $y$ and therefore it is a subset of ${B}_{r}$

•
A nonmetric topology would be the finite complement^{} topology on infinite sets^{}, in which a set is called open, if its complement is finite.
Title  open set 

Canonical name  OpenSet 
Date of creation  20130322 12:39:25 
Last modified on  20130322 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 