For example, the set of all open intervals in the real number line
can be written as a union of some family of open intervals.
[1] Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets.
Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on
A weaker notion related to bases is that of a subbase for a topology.
For the real line, the collection of all open intervals is a base for the topology.
So is the collection of all open intervals with rational endpoints, or the collection of all open intervals with irrational endpoints, for example.
Note that two different bases need not have any basic open set in common.
is the minimum cardinality of a base for its topology, called the weight of
From the examples above, the real line has countable weight.
can also be characterized as the set of all arbitrary unions of finite intersections of elements of
or is empty, then condition (B2) is automatically satisfied (by taking
For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set.
But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
is only a subbase for the topology, not a base: a finite open interval
A non-empty family of subsets of a set X that is closed under finite intersections of two or more sets, which is called a π-system on X, is necessarily a base for a topology on X if and only if it covers X.
But nevertheless, many topologies are defined by bases that are also closed under finite intersections.
is closed under finite intersections and so each forms a basis for some topology on
: The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties.
Closed sets are equally adept at describing the topology of a space.
There is, therefore, a dual notion of a base for the closed sets of a topological space.
satisfying these properties forms a base for the closed sets of a topology on
The closed sets of this topology are precisely the intersections of members of
In some cases it is more convenient to use a base for the closed sets rather than the open ones.
We shall work with notions established in (Engelking 1989, p. 12, pp.
Note that, unlike a basis, the sets in a network need not be open.
We define the weight, w(X), as the minimum cardinality of a basis; we define the network weight, nw(X), as the minimum cardinality of a network; the character of a point,
(since compact metrizable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrizable exactly in case they are second countable.
(An application of this, for instance, is that every path in a Hausdorff space is compact metrizable.)
Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length ≥ κ+.