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 , which satisfy
If and and then (every set containing a neighborhood of is a neighborhood of itself).
If then .
For every there is a , such that and for every .
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 , having the following properties:
Any union of open sets is open.
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.
The open ball around is open. Indeed, for every the open ball with radius around y is a subset of , because for every within this ball we have:
So and thus is in . This holds for every in the ball around and therefore it is a subset of
|Date of creation||2013-03-22 12:39:25|
|Last modified on||2013-03-22 12:39:25|
|Last modified by||mathwizard (128)|