An element of a topology, a mathematical construct that turns a set into a topological space. All theorems of topology are ultimately derived from the properties of open sets in given topologies.
The above writeups imply that this concept applies only to metric spaces. This is incorrect; it applies to topological spaces in general. Otherwise, it would be like saying "Green is a color that Oldsmobiles are painted."
In every topology defined on a given set X, the set
X itself is open, as is the empty set. Other open
sets are subsets of X, determined by a predicate or formula
that characterizes the space, under certain restrictions. The restrictions
are more rigorously stated in topology, but roughly, (finite) intersections and (arbitrary)
unions of open sets are also open in the topology.
An open set about (containing) a particular element (point) of the space
is called a neighborhood of that point.
For every open set u in a space X, X
- u is a closed set. Although the Kuratowski closure
theorem shows that we could just as easily define a topology based upon
closed sets, it is conventional to use open sets instead.
The term 'open' stems from the origins of topology, which sought to
explain the properties of the real number line. The "Euclidean"
topology E is determined on the set of real numbers R
when we assert that all open intervals of the real line are open in it
(defining an open interval u for each r e R
and e > 0 such that
{x ∈ u <-> (r - e < x
< r + e)}).
"Open set" is frequently defined in terms of metric spaces, but this
is a bad idea, as it confuses the concept of a metric space with that of
a topological space. The confusion arises from the fact that for
every metric space, there is a derived metric topology whose most basic
open sets are all points less than a certain distance to a particular point.
So, in a metric space, "open sets" (or "open balls" since they resemble
balls in E3) are the open sets of its metric
topology.