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''of a topological space $X$, base of a topology, basis of a topology, open base''
 
''of a topological space $X$, base of a topology, basis of a topology, open base''
  
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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''.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530032.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530033.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530035.png" />-local if each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530036.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530037.png" /> intersecting with at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530038.png" /> elements of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530039.png" />; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530040.png" />, the base is referred to as discrete; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530041.png" /> is finite, it is called bounded locally finite; and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530042.png" />, it is called locally countable. A base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530043.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530045.png" />-point base (or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530047.png" />-local base) if it is a union of a set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530049.png" />-point (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530050.png" />-local) bases; examples are, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530053.png" />-disjoint, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530055.png" />-point finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530057.png" />-discrete and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530059.png" />-locally finite bases.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530060.png" />-discrete or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530061.png" />-locally finite base is metrizable (the converse proposition is true in the former case only).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530062.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530063.png" /> is called uniform (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530065.png" />-uniform) if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530066.png" /> (each compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530067.png" />) and for each one of the neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530068.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530069.png" />) only a finite number of elements of the base contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530070.png" /> (intersect with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530071.png" />) and at the same time intersect with the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530072.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530073.png" />). A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530074.png" /> is metrizable if and only if it is paracompact with a uniform base (a Kolmogorov or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530075.png" />-space with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530076.png" />-uniform base).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530077.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530078.png" /> is called regular if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530079.png" /> and an arbitrary neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530080.png" /> of it there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530081.png" /> such that the set of all the elements of the base which intersect both with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530083.png" /> is finite. An accessible or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530084.png" />-space is metrizable if and only if it has a regular 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530086.png" />-base (lattice base), which is a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530087.png" /> of open sets in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530088.png" /> such that each non-empty open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530089.png" /> contains a non-empty set from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530090.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530091.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530092.png" /> according to Hausdorff. All bases are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530093.png" />-bases, but the converse is not true; thus, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530094.png" /> in the [[Stone–Čech compactification|Stone–Čech compactification]] of the set of natural numbers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530095.png" /> forms only a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530096.png" />-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|Stone–Čech compactification]] of the set of natural numbers in $  \mathbf Z  ^ {+} $
 +
forms only a $  \pi $-
 +
base.
  
 
====References====
 
====References====
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<TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR>
 
<TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR>
 
</table>
 
</table>
 
 
  
 
====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530097.png" />) is called point finite if every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530098.png" /> belongs to finitely many members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b01530099.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b015300100.png" /> is finite for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b015300101.png" />. Note that the families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b015300102.png" /> can have arbitrary large finite cardinalities, in contrast to the definition of bounded point finiteness, when the cardinalities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b015300103.png" /> are bounded by a fixed finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015300/b015300104.png" />. Similar remarks apply to local finiteness.
+
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.
  
 
{{TEX|part}}
 
{{TEX|part}}

Revision as of 12:25, 17 March 2020


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.

How to Cite This Entry:
Base. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Base&oldid=44780
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article