# Base

*of a topological space $X$, base of a topology, basis of a topology, open base*

A family $\mathfrak{B}$ of open subsets of $X$ such that each open subset $G \subseteq X$ is a union of subcollections $U \subseteq \mathfrak{B}$. The concept of a base is a fundamental concept in topology: in many problems concerned with open sets of some space it is sufficient to restrict the considerations to its base. A space can have many bases, the largest one of which is the family of all open sets. The minimum of the cardinalities of all bases is called the *weight* of the topological space $X$. In a space of weight $\tau$ there exists an everywhere-dense set of cardinality $\le \tau$. Spaces with a countable base are also referred to as spaces satisfying the second axiom of countability. The dual concept of a closed base, formed by the complements of the elements of a base, is used in compactification theory.

A local base of a space $X$ at a point $x \in X$ (a base of the point $x$) is a family $\mathfrak{B}(x)$ of open sets of $X$ with the following property: For any neighbourhood $O_x$ of $x$ it is possible to find an element $V \in \mathfrak{B}(x)$ such that $x \in V \subseteq O_x$. Spaces with a countable local base at every point are also referred to as spaces satisfying the first axiom of countability. A family $\mathfrak{B}$ of open sets in $X$ is a base if and only if it is a local base of each one of its points $x \in X$.

Let $\mathfrak{m}, \mathfrak{n}$ be cardinal numbers. A base $\mathfrak{B}$ of the space $X$ is called an $\mathfrak{m}$-point base if each point $x \in X$ belongs to at most $\mathfrak{m}$ elements of the family $\mathfrak{B}$; in particular, if $\mathfrak{m} = 1$, the base is called *disjoint*; if $\mathfrak{m}$ is finite, it is called *bounded point finite*; and if $\mathfrak{m} = \aleph_0$, it is called *point countable*.

A base $ \mathfrak B $ of the space $ X $ is called $ \mathfrak m $- local if each point $ x \in X $ has a neighbourhood $ O _ {x} $ intersecting with at most $ \mathfrak m $ elements of the family $ \mathfrak B $; in particular, if $ \mathfrak m = 1 $, the base is referred to as discrete; if $ \mathfrak m $ is finite, it is called bounded locally finite; and if $ \mathfrak m = \aleph _ {0} $, it is called locally countable. A base $ \mathfrak B $ is called an $ ( \mathfrak n - \mathfrak m ) $- point base (or an $ ( \mathfrak n - \mathfrak m ) $- local base) if it is a union of a set of cardinality $ \mathfrak n $ of $ \mathfrak m $- point ( $ \mathfrak m $- local) bases; examples are, for $ \mathfrak n = \aleph _ {0} $, $ \sigma $- disjoint, $ \sigma $- point finite, $ \sigma $- discrete and $ \sigma $- locally finite bases.

These concepts are used mainly in the criteria of metrizable spaces. Thus, a regular space with a countable base, or satisfying the first axiom of countability and with a point countable base, is metrizable; a regular space with a $ \sigma $- discrete or $ \sigma $- locally finite base is metrizable (the converse proposition is true in the former case only).

A base $ \mathfrak B $ of the space $ X $ is called uniform ( $ k $- uniform) if for each point $ x \in X $( each compact subset $ F $) and for each one of the neighbourhoods $ O _ {x} $( $ O _ {F} $) only a finite number of elements of the base contain $ x $( intersect with $ F $) and at the same time intersect with the complement $ X \setminus O _ {x} $( $ X \setminus O _ {F} $). A space $ X $ is metrizable if and only if it is paracompact with a uniform base (a Kolmogorov or $ T _ {0} $- space with a $ k $- uniform base).

A base $ \mathfrak B $ of the space $ X $ is called regular if for each point $ x \in X $ and an arbitrary neighbourhood $ O _ {x} $ of it there exists a neighbourhood $ O _ {x} ^ \prime $ such that the set of all the elements of the base which intersect both with $ O _ {x} ^ \prime $ and $ X\setminus O _ {x} $ is finite. An accessible or $ T _ {1} $- space is metrizable if and only if it has a regular base.

A generalization of the concept of a base is the so-called $ \pi $- base (lattice base), which is a family $ \mathfrak B $ of open sets in the space $ X $ such that each non-empty open set in $ X $ contains a non-empty set from $ \mathfrak B $, i.e. $ \mathfrak B $ is dense in $ X $ according to Hausdorff. All bases are $ \pi $- bases, but the converse is not true; thus, the set $ \mathbf Z ^ {+} $ in the Stone–Čech compactification of the set of natural numbers in $ \mathbf Z ^ {+} $ forms only a $ \pi $- base.

#### References

[1] | P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |

[2] | P.S. [P.S. Uryson] Urysohn, , Works on topology and other fields of mathematics , 1–2 , Leningrad (1951) (In Russian) |

[3] | P.S. Aleksandrov, B.A. Pasynkov, "An introduction to the theory of topological spaces and general dimension theory" , Moscow (1973) (In Russian) |

[4] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |

[5] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |

#### Comments

Besides the notions of a bounded point-finite base and a bounded local-finite base one also uses that of a point-finite base and a local-finite base. A base (or any family of subsets $ \mathfrak B $) is called point finite if every point $ x $ belongs to finitely many members of $ \mathfrak B $, i.e. if $ \mathfrak B _ {x} = \{ {B \in \mathfrak B } : {x \in B } \} $ is finite for every $ x $. Note that the families $ \mathfrak B _ {x} $ can have arbitrary large finite cardinalities, in contrast to the definition of bounded point finiteness, when the cardinalities of $ \mathfrak B _ {x} $ are bounded by a fixed finite $ \mathfrak m $. Similar remarks apply to local finiteness.

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