In a metric space, a set U is an open subset of X if for every point x in the set, there exists some positive ε such that B(x,ε)={x in X | |x|;<ε} is contained in U.
For example, in the real numbers the set (0,1) = {numbers between 0 and 1, not including either} is open, whereas [0,1] = {numbers between 0 and 1, including both} is not (ie. because no such positive ε exists for 1 or 0).

This definition gives rise the properties that:

...which become the defining properties of open sets in topological spaces.
Let's say we have a metric space, consisting of a set of points X, and a function d(x,y) which gives the distance between any two points x and y which are members of X.

A subset V of X is said to be an open set where for any member, v, of V, all the members of X which have a distance from v less than some positive nonzero constant are also members of V

That is:

For each v in V, there is some constant, C > 0 such that for all x in X, if d(vx) < C, then x is also in V.
Or alternatively:
for all v in V, there exists r > 0 such that BX(v, r) is a subset of V
(where BX(v, r) is an open ball in X with radius r and centered on v.)

A closed set in X is just the complement in X of some open set V which is a subset of X.

By convention, the empty set is considered to be open. (We need convention because there are no members of the empty set to apply the condition to.)

X itself is trivially an open set, and, as the complement of the empty set, it is also a closed set.

If we take X as the reals, under the usual topoogy, then the set [0,1] (all the reals between and including 0 and 1) is not open. In a sense, this is because 0 and 1, the greatest lower bound (infimum) and least upper bound (supremum) respectively, are both members of [0,1], which contradicts our requirement for the constant C to be nonzero. In the case of the set (0,1) (just the same as [0,1], except it doesn't contain 0 and 1), we can use the fact that 0 and 1 are not in the set to construct a nonzero C for any point inside the set.

Any subset of X constructed by union or (finite) intersection operations on these open sets is also open, so that you can never get to a set that isn't open by doing unions or (a finite number of) intersections on these sets.

This set of open subsets of X defines the metric topology on X.

If we have other ways of specifying subsets of X which meet this requirement of closure under union and finite intersection, they will define different topologies on it. That is, a way of specifying the open sets is pretty much all you need to define a topology on your collection of points, X. The concept of a topological space is far more general than that of a metric space - it doesn't need to have any distance function that makes sense, for example. Non-metric (or even weirder) spaces may be harder to visualise, since they correspond less well to our normal physical intuitions about "space".

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.

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