Open set

These conditions are very loose, and allow enormous flexibility in the choice of open sets.

[1] In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined.

The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general.

Intuitively, an open set provides a method to distinguish two points.

In this manner, one may speak of whether two points, or more generally two subsets, of a topological space are "near" without concretely defining a distance.

Note that ε > 0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy.

Clearly, these points approximate x to a greater degree of accuracy than when ε = 1.

The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller.

In particular, sets of the form (−ε, ε) give us a lot of information about points close to x = 0.

Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x.

This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−ε, ε)), one may find different results regarding the distance between 0 and other real numbers.

For example, if we were to define R as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis; a member of this neighborhood basis is referred to as an open set.

In fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers.

Of course, this collection would have to satisfy certain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance.

Once we begin to define "smaller" sets containing x, we tend to approximate x to a greater degree of accuracy.

Bearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy.

satisfying d(x, y) < ε belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U.

This notion is relative to a containing set and a specific topology on it.

As a concrete example of this, if U is defined as the set of rational numbers in the interval

Every subset A of a topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A.

It can be constructed by taking the union of all the open sets contained in A.

The open sets of the usual Euclidean topology of the real line

For a more advanced example reminiscent of the discrete topology, suppose that

regular closed set) is an open subset (resp.

is a closed subset) although in general,[note 1] the converses are not true.

the following may be deduced: Moreover, a subset is a regular open set if and only if it is preopen and semi-closed.

semi-open, preopen, b-open) set is a semi-preopen (resp.

α-open, b-open, semi-preopen) sets are once again preopen (resp.

Moreover, the space is totally disconnected if and only if the closure of every preopen subset is open.