Simply_connected
Simply connected space
Space which has no holes through it
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected[1]) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
A topological space is called simply connected if it is path-connected and any loop in
defined by
can be contracted to a point: there exists a continuous map
such that
restricted to
is
Here,
and
denotes the unit circle and closed unit disk in the Euclidean plane respectively.
An equivalent formulation is this: is simply connected if and only if it is path-connected, and whenever
and
are two paths (that is, continuous maps) with the same start and endpoint (
and
), then
can be continuously deformed into
while keeping both endpoints fixed. Explicitly, there exists a homotopy
such that
and
A topological space is simply connected if and only if
is path-connected and the fundamental group of
at each point is trivial, i.e. consists only of the identity element. Similarly,
is simply connected if and only if for all points
the set of morphisms
in the fundamental groupoid of
has only one element.[2]
In complex analysis: an open subset is simply connected if and only if both
and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes an example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. A relaxation of the requirement that
be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components is simply connected.
Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.
The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility.
- The Euclidean plane
is simply connected, but
minus the origin
is not. If
then both
and
minus the origin are simply connected.
- Analogously: the n-dimensional sphere
is simply connected if and only if
- Every convex subset of
is simply connected.
- A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are not simply connected.
- Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
- For
the special orthogonal group
is not simply connected and the special unitary group
is simply connected.
- The one-point compactification of
is not simply connected (even though
is simply connected).
- The long line
is simply connected, but its compactification, the extended long line
is not (since it is not even path connected).
A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus (the number of handles of the surface) is 0.
A universal cover of any (suitable) space is a simply connected space which maps to
via a covering map.
If and
are homotopy equivalent and
is simply connected, then so is
The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is which is not simply connected.
The notion of simple connectedness is important in complex analysis because of the following facts:
- The Cauchy's integral theorem states that if
is a simply connected open subset of the complex plane
and
is a holomorphic function, then
has an antiderivative
on
and the value of every line integral in
with integrand
depends only on the end points
and
of the path, and can be computed as
The integral thus does not depend on the particular path connecting
and
- The Riemann mapping theorem states that any non-empty open simply connected subset of
(except for
itself) is conformally equivalent to the unit disk.
The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.
- Fundamental group – Mathematical group of the homotopy classes of loops in a topological space
- Deformation retract – Continuous, position-preserving mapping from a topological space into a subspacePages displaying short descriptions of redirect targets
- n-connected space
- Path-connected – Topological space that is connectedPages displaying short descriptions of redirect targets
- Unicoherent space
- "n-connected space in nLab". ncatlab.org. Retrieved 2017-09-17.
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- Conway, John (1986). Functions of One Complex Variable I. Springer. ISBN 0-387-90328-3.
- Bourbaki, Nicolas (2005). Lie Groups and Lie Algebras. Springer. ISBN 3-540-43405-4.
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