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Let  $  \mathbf n = \langle  n _ {i} \rangle _ {i} $
be a sequence of positive integers. From  $  \mathbf n $
one constructs a topological space as follows.

Let  $  T _ {0} $
be a torus in  $  \mathbf R  ^ {3} $; 
inside  $  T _ {0} $
one takes a torus  $  T _ {1} $
wrapped around longitudinally  $  n _ {1} $
times, in a smooth fashion without folding back; inside  $  T _ {1} $
one takes a torus  $  T _ {2} $
wrapped around  $  n _ {2} $
times in the same way. Continuing this procedure indefinitely, one obtains a decreasing sequence of tori. Its intersection is called the  $  \mathbf n $-
adic solenoid  $  \Sigma _ {\mathbf n }  $.

The basic properties of  $  \Sigma _ {\mathbf n }  $
are that it is a one-dimensional [[Continuum|continuum]] which, moreover, is indecomposable (cf. [[Indecomposable continuum|Indecomposable continuum]]).

$  \Sigma _ {\mathbf n }  $
is also a [[Topological group|topological group]]; this can be seen if one considers an alternative construction of  $  \Sigma _ {\mathbf n }  $
as the inverse limit of the following inverse sequence:

$$ 
{} \dots S _ {3}  \rightarrow ^ { {f _ 3} }  S _ {2}   \mathop \rightarrow \limits ^ { {f _ {2} }}   S _ {1}  \rightarrow ^ { {f _ 1} }  S _ {0} ,
$$

where each  $  S _ {i} $
is the unit circle and  $  f _ {i} :  S _ {i} \rightarrow S _ {i-} 1 $
is defined by  $  f _ {i} ( z)= z ^ {n _ {i} } $. 
There are various other ways in which one can construct the solenoids, see, e.g., [[#References|[a3]]].

Solenoids were first defined by L. Vietoris [[#References|[a2]]] (for the sequence  $  \langle  2, 2 ,\dots \rangle $) 
and by D. van Dantzig [[#References|[a1]]] (for all constant sequences).

Solenoids are also important in [[Topological dynamics|topological dynamics]]; on them one can define a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] structure [[#References|[a4]]] which has a locally disconnected minimal set of almost-periodic motions.

There is a complete classification of the solenoids: first, without loss of generality one may assume that the numbers  $  n _ {i} $
are prime. Call two sequences of primes  $  \mathbf p $
and  $  \mathbf q $
equivalent if one can delete from each a finite number of terms such that in the reduced sequences  $  \mathbf p  ^  \prime  $
and  $  \mathbf q  ^  \prime  $
every prime is counted the same number of times. One can then show that  $  \Sigma _ {\mathbf p }  $
and  $  \Sigma _ {\mathbf q }  $
are homeomorphic if and only if  $  \mathbf p $
and  $  \mathbf q $
are equivalent. See [[#References|[a5]]] and [[#References|[a6]]].

Finally one can characterize the solenoids as those metric continua that are homogeneous and have the property that every proper subcontinuum is an [[Arc (topology)|arc]]. See [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. van Dantzig,   "Ueber topologisch homogene Kontinua"  ''Fund. Math.'' , '''15'''  (1930)  pp. 102–125</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Vietoris,   "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen"  ''Math. Ann.'' , '''97'''  (1927)  pp. 454–472</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.H. Bing,   "A simple closed curve is the only homogeneous bounded plane continuum that contains an arc"  ''Canad. Math. J.'' , '''12'''  (1960)  pp. 209–230</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.C. McCord,   "Inverse limit sequences with covering maps"  ''Trans. Amer. Math. Soc.'' , '''114'''  (1965)  pp. 197–209</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C.L. Hagopian,   "A characterization of solenoids"  ''Pacific J. Math.'' , '''68'''  (1977)  pp. 425–435</TD></TR></table>