# 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 of the space is called -local if each point has a neighbourhood intersecting with at most elements of the family ; in particular, if , the base is referred to as discrete; if is finite, it is called bounded locally finite; and if , it is called locally countable. A base is called an -point base (or an -local base) if it is a union of a set of cardinality of -point (-local) bases; examples are, for , -disjoint, -point finite, -discrete and -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 -discrete or -locally finite base is metrizable (the converse proposition is true in the former case only).

A base of the space is called uniform (-uniform) if for each point (each compact subset ) and for each one of the neighbourhoods () only a finite number of elements of the base contain (intersect with ) and at the same time intersect with the complement (). A space is metrizable if and only if it is paracompact with a uniform base (a Kolmogorov or -space with a -uniform base).

A base of the space is called regular if for each point and an arbitrary neighbourhood of it there exists a neighbourhood such that the set of all the elements of the base which intersect both with and is finite. An accessible or -space is metrizable if and only if it has a regular base.

A generalization of the concept of a base is the so-called -base (lattice base), which is a family of open sets in the space such that each non-empty open set in contains a non-empty set from , i.e. is dense in according to Hausdorff. All bases are -bases, but the converse is not true; thus, the set in the Stone–Čech compactification of the set of natural numbers in forms only a -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 ) is called point finite if every point belongs to finitely many members of , i.e. if is finite for every . Note that the families can have arbitrary large finite cardinalities, in contrast to the definition of bounded point finiteness, when the cardinalities of are bounded by a fixed finite . Similar remarks apply to local finiteness.

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