A
*solenoid*
is a continuum homeomorphic to
the inverse limit
of the inverse sequence of unit circles in the complex
plane with bonding maps
, where
is a sequence of prime numbers; it is
called a
*-adic solenoid*
. The solenoid
is known as a
*dyadic solenoid*
.

Geometrically, solenoid can be described as the intersection of a sequence of solid tori such that wraps times around without folding and is -thin, for each , where . See Figure A.

- Each solenoid can be constructed (up to a homeomorphism)
as the quotient space of the product by identifying
each point with , where is
a homeomorphism of the Cantor set such that
for every
there exist a closed-open subset of
and a positive integer such that
is a cover of consisting of
pairwise disjoint subsets of with diameters less than
[Gutek 1980].
- Each solenoid
is an Abelian
topological group with a group operation
and the neutral element
.
- Either of the following conditions is equivalent for a
nondegenerate continuum different from
a simple closed curve to be a solenoid.
- is homeomorhic to a one-dimensional
topological group [Hewitt 1963];
- is indecomposable and is homeomorphic to a topological group
[Chigogidze 1996, Theorem 8.6.18];
- is circle-like, has the property of Kelley and contains no local end point [Krupski 1984c, Theorem (4.3)];
- is circle-like, has the property of Kelley, each proper
nondegenerate subcontinuum of is an arc and has no
end pointsend point;
- is circle-like, has the property of Kelley and has an
open cover by Cantor bundles of open arcs (i.e., sets homeomorphic
to the product
of the Cantor set and the
open interval ) [Krupski 1982];
- is homogeneous, contains no proper,
nondegenerate, terminal subcontinua
and sufficiently small subcontinua of are not
-ods [Krupski 1995, Theorem
3.1];
- is a homogeneous curve containing an open subset
such that some component of does not have the
disjoint arcs property [Krupski 1995, p.
166];
- is a homogeneous finitely cyclic (or, equivalently, -junctioned) curve that is not tree-like and contains no
nondegenerate, proper, terminal subcontinua
[Krupski et al. XXXXb], [Duda et al. 1991].
- is openly homogeneous and
sufficiently small subcontinua of are arcs
[Prajs 1989];

- is homeomorhic to a one-dimensional
topological group [Hewitt 1963];
- Solenoid
is a continuous image of
if and only if the sequence
is a
factorant of sequence
, i.e.,
there exists such that for each there is such
that
is a factor of
.
Two solenoids are homeomorphic if and only if each of them is a continuous image of another [Cook 1967], [D. van Dantzig 1930, Satz 8, p. 122].

There is a family of solenoids of cardinality such that no member of the family is a continuous image of another.

- Each monotonemonotone map image of a solenoid is
homeomorphic to [Krupski 1984b, Theorem 5].
Each open map transforms onto a solenoid or onto an arc-like continuum with the property of Kelley and with arcs as proper nondegenerate subcontinua; if the map is a local homeomorphism, then its image is a solenoid [Krupski 1984a].

- The composant of a solenoid containing is a one-parameter
topological subgroup of , i.e. it is a one-to-one continuous
homomorphic image of the additive group of the reals.
- Any two composants of any two solenoids are
homeomorphic [R. de Man 1995].
- No solenoid can be mapped onto a strongly self-entwined continuum. In
particular, it cannot be mapped onto a circle-like plane
continuum which is a common part of a descending sequence of
circular chains such that circles times
in clockwisely and then times
counter-clockwisely and the first link of contains the
closure of the first link of [Rogers 1971b].
- No movable continuum (in particular no
continuum lying in a surface or a tree-like continuum) can be continuously mapped onto a
solenoid. Alternatively, if the first Alexander-Cech
cohomology group of a continuum is finitely
divisiblefinitely divisible group, then cannot be
mapped onto a solenoid [Krasinkiewicz 1976, Remark, p. 46, 4.1, 4.9.,
5.1], [Krasinkiewicz 1978, Corollary
7.3], [Rogers 1975].
- Every nonplanar, circle-like continuum has the
shape of a solenoid [Krasinkiewicz 1976, remark, p.
46]. Two solenoids have the same shape
if and only if they are homeomorphic
[Godlewski 1970].
- Any autohomeomorphism of
is isotopic
to a homeomorphism which is induced by a map
of the inverse sequences which
define
( can be a group translation,
the involution, a power map or its inverse, or compositions
of these maps). Maps and have equal the topological
entropies and are
semi-conjugate if the entropy is positive
[Kwapisz 2001, Theorems 1-3, pp. 252-253], [D. van Dantzig 1930, Satz
9, p. 125].
The topological group of all autohomeomorphisms (with the compact-open topology) of a solenoid is homeomorphic (but not isomorphic) to the the product , where is the Hilbert space and the group of all topological group automorphisms of is equipped with the discrete topology and it is equal to , or , or [Keesling 1972, Theorems 3.1 and 2.4].

- If the spaces of all autohomeomorphisms of two solenoids are homeomorphic,
then the solenoids are isomorphic as topological groups
[Keesling 1972, Corollary 3.9].
- Any map
is,
for every
, -homotopic to a map
induced by a map
between inverse sequences
defining the corresponding solenoids [Rogers et al. 1971].
- A -adic solenoid admits a mean if and
only if infinitely many numbers in the sequence
equal 2 [Krupski XXXXa]. The same condition is
equivalent to the non-existence of exactly 2-to-1 map
defined on the solenoid [Debski 1992].
- The hyperspace of all subcontinua of any
solenoid is homeomorphic the cone over
[Rogers 1971a], [Nadler 1991, p. 202].
- The family of all solenoids in the cube
(as a subset of the hyperspace ) is Borel
and not
[Krupski XXXXc].