PlanetMath: open set

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open set

(Definition)

In a metric space a set is called an open subset of or just open, if for every $x\in O$ there is an open ball around such that $S\subset O$ . If is the distance from to then the open ball with radius around is given as:

$\displaystyle B_r=\{y\in M\vert d(x,y)<r\}.$

Using the idea of an open ball one can define a neighborhood of a point . A set containing is called a neighborhood of if there is an open ball around 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 with a set of neighborhoods of called for every $x\in X$ , which satisfy

$x\in U$ for every $U\in U_x$
If $U\in U_x$ and $V\subset X$ and $U\subset V$ then $V\in U_x$ (every set containing a neighborhood of is a neighborhood of itself).
If $U,V\in U_x$ then $U\cap V\in U_x$ .
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 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 and a set of subsets of , which we call open sets, called $\mathcal{O}$ , having the following properties:

$\emptyset\in\mathcal{O}$ and $X\in\mathcal{O}$ .
Any union of open sets is open.
Finite 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 is open because for every $a\in I$ the open ball with radius $\min(a,1-a)$ is always a subset of . (Using the standard metric $d(x,y)=\vert x-y\vert$ .)
The open ball around is open. Indeed, for every $y\in B_r$ the open ball with radius around y is a subset of , because for every within this ball we have:
$\displaystyle d(x,z)\leq d(x,y)+d(y,z)<d(x,y)+r-d(x,y)=r.$
So and thus is in . This holds for every in the ball around and therefore it is a subset of
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.