Wikipedia / Simply connected space
Wikipedia / Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) 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.
Definition and equivalent formulations
A topological space is called if it is path-connected and any loop denotes the unit circle and closed unit disk in the Euclidean plane respectively.
while keeping both endpoints fixed. Explicitly, there exists a homotopy
is path-connected and the fundamental group at each point is trivial, i.e. consists only of the identity element the set of morphisms in the fundamental groupoid has only one element.
In complex analysis and its complement in the Riemann sphere 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.
Informal discussion
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 dimension, is called contractibility.
Examples
The Euclidean plane minus the origin are simply connected.
Analogously: the n-dimensional sphere
Every convex subset 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.
the special orthogonal group is not simply connected and the special unitary group is simply connected.
is simply connected).
The long line is not (since it is not even path connected).
Properties
A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus (the number of of the surface) is 0.
via a covering map.
which is not simply connected.
The notion of simple connectedness is important in complex analysis because of the following facts:
Cauchy's integral theorem is a simply connected open subset of the complex plane is a holomorphic function has an antiderivative and the value of every line integral
The Riemann mapping theorem itself) is conformally equivalent to the unit disk.
The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.
See also
References