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.
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In topology, a topological space is called simplyconnected (or 1-connected, or 1-simplyconnected) if it is path-connected and every path between two...
Connected and disconnected subspaces of R² In topology and related branches of mathematics, a connectedspace is a topological space that cannot be represented...
a locally simplyconnectedspace is a topological space that admits a basis of simplyconnected sets. Every locally simplyconnectedspace is also locally...
topological space X is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space X is...
guarantee its existence: Let X {\displaystyle X} be a connected, locally simplyconnected topological space; then, there exists a universal covering p : X ~...
orbit space for this action is RPn. This action is actually a covering space action giving Sn as a double cover of RPn. Since Sn is simplyconnected for...
is currently unknown whether the universe is simplyconnected like euclidean space or multiply connected like a torus. To date, no compelling evidence...
{\displaystyle \mathbf {v} } is defined is not a simplyconnected open space. Say again, in a simplyconnected open region, an irrotational vector field v...
_{1}(S^{1})\cong \mathbb {Z} .} Any path connected, locally path connected and locally simplyconnected topological space X admits a universal covering. An abstract...
makes certain calculations much easier. Rational homotopy types of simplyconnectedspaces can be identified with (isomorphism classes of) certain algebraic...
metric space is a metric space where any two points are the endpoints of a minimizing geodesic. Hadamard space is a complete simplyconnectedspace with...
consistent with the universe having infinite extent and being a simplyconnectedspace, though cosmological horizons limit our ability to distinguish between...
irreducible simplyconnected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces). The irreducible simplyconnected symmetric...
contractible space is path connected and simplyconnected. Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all...
′ ( 1 ) . {\displaystyle -\gamma '(1).} The simply-connectedspace forms (the n-sphere, hyperbolic space, and R n {\displaystyle \mathbb {R} ^{n}} ) are...
In topology, a branch of mathematics, a topological space X is said to be simplyconnected at infinity if for any compact subset C of X, there is a compact...
each space is simplyconnected. However, the two spaces cannot be homeomorphic because deleting a point from R2 leaves a non-simply-connectedspace but...
In mathematics, hyperbolic space of dimension n is the unique simplyconnected, n-dimensional Riemannian manifold of constant sectional curvature equal...
general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is...
connected complete Riemannian manifold with constant curvature, but is not assumed to be simply-connected, then consider the universal covering space...
transformations of the fundamental group of a locally simplyconnectedspace on an covering space is wandering and free. Such actions can be characterized...
manifold which is timelike multiply connected has a diffeomorphic universal covering space which is timelike simplyconnected. For instance, a three-sphere...
Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simplyconnected Riemannian...
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