In mathematics, a base (or basis; pl.: bases) for the topology τ of a topological space (X, τ) is a family ${\displaystyle {\mathcal {B}}}$ of open subsets of X such that every open set of the topology is equal to the union of some sub-family of ${\displaystyle {\mathcal {B}}}$. For example, the set of all open intervals in the real number line ${\displaystyle \mathbb {R} }$ is a basis for the Euclidean topology on ${\displaystyle \mathbb {R} }$ because every open interval is an open set, and also every open subset of ${\displaystyle \mathbb {R} }$ can be written as a union of some family of open intervals.

Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets.^[1] Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.

Not all families of subsets of a set ${\displaystyle X}$ form a base for a topology on ${\displaystyle X}$. Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on ${\displaystyle X}$, obtained by taking all possible unions of subfamilies. Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a subbase for a topology. Bases for topologies are also closely related to neighborhood bases.

Definition and basic properties

Given a topological space ${\displaystyle (X,\tau )}$, a base^[2] (or basis^[3]) for the topology ${\displaystyle \tau }$ (also called a base for ${\displaystyle X}$ if the topology is understood) is a family ${\displaystyle {\mathcal {B}}\subseteq \tau }$ of open sets such that every open set of the topology can be represented as the union of some subfamily of ${\displaystyle {\mathcal {B}}}$.^{[note 1]} The elements of ${\displaystyle {\mathcal {B}}}$ are called basic open sets. Equivalently, a family ${\displaystyle {\mathcal {B}}}$ of subsets of ${\displaystyle X}$ is a base for the topology ${\displaystyle \tau }$ if and only if ${\displaystyle {\mathcal {B}}\subseteq \tau }$ and for every open set ${\displaystyle U}$ in ${\displaystyle X}$ and point ${\displaystyle x\in U}$ there is some basic open set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle x\in B\subseteq U}$.

For example, the collection of all open intervals in the real line forms a base for the standard topology on the real numbers. More generally, in a metric space ${\displaystyle M}$ the collection of all open balls about points of ${\displaystyle M}$ forms a base for the topology.

In general, a topological space ${\displaystyle (X,\tau )}$ can have many bases. The whole topology ${\displaystyle \tau }$ is always a base for itself (that is, ${\displaystyle \tau }$ is a base for ${\displaystyle \tau }$). 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. One of the topological properties of a space ${\displaystyle X}$ is the minimum cardinality of a base for its topology, called the weight of ${\displaystyle X}$ and denoted ${\displaystyle w(X)}$. From the examples above, the real line has countable weight.

If ${\displaystyle {\mathcal {B}}}$ is a base for the topology ${\displaystyle \tau }$ of a space ${\displaystyle X}$, it satisfies the following properties:^[4]

(B1) The elements of ${\displaystyle {\mathcal {B}}}$ cover ${\displaystyle X}$, i.e., every point ${\displaystyle x\in X}$ belongs to some element of ${\displaystyle {\mathcal {B}}}$.

(B2) For every ${\displaystyle B_{1},B_{2}\in {\mathcal {B}}}$ and every point ${\displaystyle x\in B_{1}\cap B_{2}}$, there exists some ${\displaystyle B_{3}\in {\mathcal {B}}}$ such that ${\displaystyle x\in B_{3}\subseteq B_{1}\cap B_{2}}$.

Property (B1) corresponds to the fact that ${\displaystyle X}$ is an open set; property (B2) corresponds to the fact that ${\displaystyle B_{1}\cap B_{2}}$ is an open set.

Conversely, suppose ${\displaystyle X}$ is just a set without any topology and ${\displaystyle {\mathcal {B}}}$ is a family of subsets of ${\displaystyle X}$ satisfying properties (B1) and (B2). Then ${\displaystyle {\mathcal {B}}}$ is a base for the topology that it generates. More precisely, let ${\displaystyle \tau }$ be the family of all subsets of ${\displaystyle X}$ that are unions of subfamilies of ${\displaystyle {\mathcal {B}}.}$ Then ${\displaystyle \tau }$ is a topology on ${\displaystyle X}$ and ${\displaystyle {\mathcal {B}}}$ is a base for ${\displaystyle \tau }$.^[5] (Sketch: ${\displaystyle \tau }$ defines a topology because it is stable under arbitrary unions by construction, it is stable under finite intersections by (B2), it contains ${\displaystyle X}$ by (B1), and it contains the empty set as the union of the empty subfamily of ${\displaystyle {\mathcal {B}}}$. The family ${\displaystyle {\mathcal {B}}}$ is then a base for $$Zdroj:https://en.wikipedia.org?pojem=Base_(topology)
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