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A [[Topological group|topological group]] in which the group operations are defined only for elements sufficiently close to the identity. The introduction of local topological groups was inspired by the study of the local structure of topological groups (that is, their structure in an arbitrary small neighbourhood of the identity, see [[#References|[1]]]). The precise definition of a local topological group is as follows.
 
A [[Topological group|topological group]] in which the group operations are defined only for elements sufficiently close to the identity. The introduction of local topological groups was inspired by the study of the local structure of topological groups (that is, their structure in an arbitrary small neighbourhood of the identity, see [[#References|[1]]]). The precise definition of a local topological group is as follows.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602201.png" /> be a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602202.png" /> an element of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602204.png" /> open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602206.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602207.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l0602209.png" /> be continuous mappings. Then the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022010.png" /> is called a local topological group if the following conditions are satisfied:
+
Let $  G $
 +
be a topological space, $  e $
 +
an element of it, $  \Theta $
 +
and $  \Omega $
 +
open subsets of $  G $
 +
and $  G \times G $,  
 +
respectively, where $  e \in \Theta $,  
 +
and let $  i : \Theta \rightarrow G $
 +
and $  m : \Omega \rightarrow G $
 +
be continuous mappings. Then the system $  ( G , e , \Theta , \Omega , i , m ) $
 +
is called a local topological group if the following conditions are satisfied:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022012.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022014.png" />;
+
1) $  ( e , g ) $
 +
and $  ( g , e ) \in \Omega $
 +
for any $  g \in G $
 +
and $  m ( ( e , g ) ) = m ( ( g , e ) ) = g $;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022020.png" />;
+
2) if $  g , h , t \in G $
 +
and $  ( g , h ) $,
 +
$  ( h , t ) $,
 +
$  ( ( g , h ) , t ) $,  
 +
$  ( g , ( h , t ) ) \in \Omega $,  
 +
then $  m ( ( m ( ( g , h ) ) , t ) ) = m ( ( g , m ( ( h , t ) ) ) ) $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022022.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022024.png" />.
+
3) $  ( g , i ( g) ) $
 +
and $  ( i ( g) , g ) \in \Omega $
 +
for any $  g \in \Theta $
 +
and $  m ( ( g , i ( g) ) ) = m ( ( i ( g), g)) = e $.
  
The local topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022025.png" /> is usually denoted simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022026.png" />; the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022027.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022028.png" /> and called the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022030.png" />; the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022031.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022032.png" /> and called the inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022033.png" />; the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022034.png" /> is called the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022036.png" />, one says that the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022038.png" /> is defined; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022039.png" />, one says that an inverse element is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022040.png" />.
+
The local topological group $  ( G , e , \Theta , \Omega , i , m ) $
 +
is usually denoted simply by $  G $;  
 +
the element $  m ( ( g , h )) $
 +
is denoted by $  gh $
 +
and called the product of $  g $
 +
and $  h $;  
 +
the element $  i ( g) $
 +
is denoted by $  g  ^ {-} 1 $
 +
and called the inverse of $  g $;  
 +
the element $  e $
 +
is called the identity of $  G $.  
 +
If $  ( g , h ) \in \Omega $,  
 +
one says that the product of $  g $
 +
and $  h $
 +
is defined; if $  g \in \Theta $,  
 +
one says that an inverse element is defined for $  g $.
  
These operations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022041.png" /> (which are not defined for all elements) induce the structure of a local topological group in an arbitrary neighbourhood of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022043.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022045.png" /> be two local topological groups. A local homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022046.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022047.png" /> is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022048.png" /> of a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022049.png" /> of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022051.png" /> into a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022052.png" /> of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022054.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022055.png" /> and for any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022056.png" /> whose product is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022057.png" /> the product of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022059.png" /> is also defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022061.png" />. Two local homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022062.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022063.png" /> are said to be equivalent if they coincide in a neighbourhood of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022064.png" />. Suppose that the local homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022065.png" /> is a [[Homeomorphism|homeomorphism]] of the neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022067.png" /> and that the inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022068.png" /> is a local homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022069.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022070.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022071.png" /> is called a local isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022073.png" />. Two local topological groups between which there is a local isomorphism are said to be locally isomorphic. For example, any local topological group is locally isomorphic to an arbitrary neighbourhood of the identity of it.
+
These operations on $  G $(
 +
which are not defined for all elements) induce the structure of a local topological group in an arbitrary neighbourhood of the identity $  e $
 +
of $  G $.  
 +
Let $  G _ {1} $
 +
and $  G _ {2} $
 +
be two local topological groups. A local homomorphism of $  G _ {1} $
 +
into $  G _ {2} $
 +
is a continuous mapping $  f $
 +
of a neighbourhood $  U _ {1} $
 +
of the identity $  e _ {1} $
 +
of $  G _ {1} $
 +
into a neighbourhood $  U _ {2} $
 +
of the identity $  e _ {2} $
 +
of $  G _ {2} $
 +
such that $  f ( e _ {1} ) = e _ {2} $
 +
and for any elements $  g , h \in U _ {1} $
 +
whose product is defined in $  G _ {1} $
 +
the product of the elements $  f ( g) $
 +
and $  f ( h) $
 +
is also defined in $  G _ {2} $
 +
and $  f ( g h ) = f ( g) f ( h) $.  
 +
Two local homomorphisms of $  G _ {1} $
 +
into $  G _ {2} $
 +
are said to be equivalent if they coincide in a neighbourhood of the identity of $  G _ {1} $.  
 +
Suppose that the local homomorphism $  f $
 +
is a [[Homeomorphism|homeomorphism]] of the neighbourhoods $  U _ {1} $
 +
and $  U _ {2} $
 +
and that the inverse mapping $  f ^ { - 1 } : U _ {2} \rightarrow U _ {1} $
 +
is a local homomorphism of $  G _ {2} $
 +
to $  G _ {1} $.  
 +
Then $  f $
 +
is called a local isomorphism of $  G _ {1} $
 +
and $  G _ {2} $.  
 +
Two local topological groups between which there is a local isomorphism are said to be locally isomorphic. For example, any local topological group is locally isomorphic to an arbitrary neighbourhood of the identity of it.
  
 
As an example of a local topological group one can take any topological group (and hence any neighbourhood of the identity of it). In the theory of local topological groups the main question is to what extent this example has a general character; that is, whether any local topological group is locally isomorphic to some topological group. In the general case the answer is negative (see [[#References|[4]]]), but in the important special case of finite-dimensional local Lie groups (cf. [[Lie group, local|Lie group, local]]) it is affirmative.
 
As an example of a local topological group one can take any topological group (and hence any neighbourhood of the identity of it). In the theory of local topological groups the main question is to what extent this example has a general character; that is, whether any local topological group is locally isomorphic to some topological group. In the general case the answer is negative (see [[#References|[4]]]), but in the important special case of finite-dimensional local Lie groups (cf. [[Lie group, local|Lie group, local]]) it is affirmative.
  
Just as in the theory of topological groups, in the theory of local topological groups one can define the concepts of (local) subgroups, normal subgroups, cosets, and quotient groups. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022074.png" /> be a local topological group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022075.png" /> be a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022076.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022077.png" /> such that in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022079.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022080.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022081.png" /> is closed. Suppose also that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022082.png" /> the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022083.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022084.png" /> and that the set
+
Just as in the theory of topological groups, in the theory of local topological groups one can define the concepts of (local) subgroups, normal subgroups, cosets, and quotient groups. For example, let $  ( G , e , \Theta , \Omega , i , m ) $
 +
be a local topological group and let $  H $
 +
be a subset of $  G $
 +
containing $  e $
 +
such that in a neighbourhood $  U $
 +
of $  e $
 +
in $  G $
 +
the set $  U \cap H $
 +
is closed. Suppose also that for any $  g \in H \cap \Theta $
 +
the element $  i ( g) $
 +
belongs to $  H $
 +
and that the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022085.png" /></td> </tr></table>
+
$$
 +
\Omega _ {H}  = \
 +
\{ {( g , h ) \in \Omega \cap ( H \times H ) } : {
 +
m ( ( g , h ) ) \in H } \}
 +
$$
  
is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022086.png" /> (under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022087.png" /> is endowed with the topology induced from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022088.png" />). Then the system
+
is open in $  H \times H $(
 +
under the assumption that $  H $
 +
is endowed with the topology induced from $  G $).  
 +
Then the system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022089.png" /></td> </tr></table>
+
$$
 +
( H , e , \Theta \cap H , \Omega _ {H} , i \mid  _ {H} ,\
 +
m \mid  _ {\Omega _ {H}  } )
 +
$$
  
is a local topological group, called a local subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060220/l06022090.png" />. For the definitions of a normal subgroup, cosets with respect to a subgroup and a quotient group, see [[#References|[1]]].
+
is a local topological group, called a local subgroup of $  G $.  
 +
For the definitions of a normal subgroup, cosets with respect to a subgroup and a quotient group, see [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lie,  F. Engel,  "Theorie der Transformationsgruppen" , '''1–3''' , Leipzig  (1930)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lie,  F. Engel,  "Theorie der Transformationsgruppen" , '''1–3''' , Leipzig  (1930)</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


A topological group in which the group operations are defined only for elements sufficiently close to the identity. The introduction of local topological groups was inspired by the study of the local structure of topological groups (that is, their structure in an arbitrary small neighbourhood of the identity, see [1]). The precise definition of a local topological group is as follows.

Let $ G $ be a topological space, $ e $ an element of it, $ \Theta $ and $ \Omega $ open subsets of $ G $ and $ G \times G $, respectively, where $ e \in \Theta $, and let $ i : \Theta \rightarrow G $ and $ m : \Omega \rightarrow G $ be continuous mappings. Then the system $ ( G , e , \Theta , \Omega , i , m ) $ is called a local topological group if the following conditions are satisfied:

1) $ ( e , g ) $ and $ ( g , e ) \in \Omega $ for any $ g \in G $ and $ m ( ( e , g ) ) = m ( ( g , e ) ) = g $;

2) if $ g , h , t \in G $ and $ ( g , h ) $, $ ( h , t ) $, $ ( ( g , h ) , t ) $, $ ( g , ( h , t ) ) \in \Omega $, then $ m ( ( m ( ( g , h ) ) , t ) ) = m ( ( g , m ( ( h , t ) ) ) ) $;

3) $ ( g , i ( g) ) $ and $ ( i ( g) , g ) \in \Omega $ for any $ g \in \Theta $ and $ m ( ( g , i ( g) ) ) = m ( ( i ( g), g)) = e $.

The local topological group $ ( G , e , \Theta , \Omega , i , m ) $ is usually denoted simply by $ G $; the element $ m ( ( g , h )) $ is denoted by $ gh $ and called the product of $ g $ and $ h $; the element $ i ( g) $ is denoted by $ g ^ {-} 1 $ and called the inverse of $ g $; the element $ e $ is called the identity of $ G $. If $ ( g , h ) \in \Omega $, one says that the product of $ g $ and $ h $ is defined; if $ g \in \Theta $, one says that an inverse element is defined for $ g $.

These operations on $ G $( which are not defined for all elements) induce the structure of a local topological group in an arbitrary neighbourhood of the identity $ e $ of $ G $. Let $ G _ {1} $ and $ G _ {2} $ be two local topological groups. A local homomorphism of $ G _ {1} $ into $ G _ {2} $ is a continuous mapping $ f $ of a neighbourhood $ U _ {1} $ of the identity $ e _ {1} $ of $ G _ {1} $ into a neighbourhood $ U _ {2} $ of the identity $ e _ {2} $ of $ G _ {2} $ such that $ f ( e _ {1} ) = e _ {2} $ and for any elements $ g , h \in U _ {1} $ whose product is defined in $ G _ {1} $ the product of the elements $ f ( g) $ and $ f ( h) $ is also defined in $ G _ {2} $ and $ f ( g h ) = f ( g) f ( h) $. Two local homomorphisms of $ G _ {1} $ into $ G _ {2} $ are said to be equivalent if they coincide in a neighbourhood of the identity of $ G _ {1} $. Suppose that the local homomorphism $ f $ is a homeomorphism of the neighbourhoods $ U _ {1} $ and $ U _ {2} $ and that the inverse mapping $ f ^ { - 1 } : U _ {2} \rightarrow U _ {1} $ is a local homomorphism of $ G _ {2} $ to $ G _ {1} $. Then $ f $ is called a local isomorphism of $ G _ {1} $ and $ G _ {2} $. Two local topological groups between which there is a local isomorphism are said to be locally isomorphic. For example, any local topological group is locally isomorphic to an arbitrary neighbourhood of the identity of it.

As an example of a local topological group one can take any topological group (and hence any neighbourhood of the identity of it). In the theory of local topological groups the main question is to what extent this example has a general character; that is, whether any local topological group is locally isomorphic to some topological group. In the general case the answer is negative (see [4]), but in the important special case of finite-dimensional local Lie groups (cf. Lie group, local) it is affirmative.

Just as in the theory of topological groups, in the theory of local topological groups one can define the concepts of (local) subgroups, normal subgroups, cosets, and quotient groups. For example, let $ ( G , e , \Theta , \Omega , i , m ) $ be a local topological group and let $ H $ be a subset of $ G $ containing $ e $ such that in a neighbourhood $ U $ of $ e $ in $ G $ the set $ U \cap H $ is closed. Suppose also that for any $ g \in H \cap \Theta $ the element $ i ( g) $ belongs to $ H $ and that the set

$$ \Omega _ {H} = \ \{ {( g , h ) \in \Omega \cap ( H \times H ) } : { m ( ( g , h ) ) \in H } \} $$

is open in $ H \times H $( under the assumption that $ H $ is endowed with the topology induced from $ G $). Then the system

$$ ( H , e , \Theta \cap H , \Omega _ {H} , i \mid _ {H} ,\ m \mid _ {\Omega _ {H} } ) $$

is a local topological group, called a local subgroup of $ G $. For the definitions of a normal subgroup, cosets with respect to a subgroup and a quotient group, see [1].

References

[1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[4] S. Lie, F. Engel, "Theorie der Transformationsgruppen" , 1–3 , Leipzig (1930)
How to Cite This Entry:
Local topological group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_topological_group&oldid=47684
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article