simply connected space

The standard example of a non-semi-locally simply connected space is the Hawaiian earring.

In particular, symplectic vector fields on simply connected spaces are Hamiltonian.

Hadamard space is a complete simply connected space with nonpositive curvature.

The definition of Simply connected space follows:

The circle is an example of a locally simply connected space which is not simply connected.

This also shows that the one-point compactification of a simply connected space need not be simply connected.

Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.

Note however, that if the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space need not be connected.

A rational space is a simply connected space all of whose homotopy groups are vector spaces over the rational numbers.

A simply connected space cannot be homeomorphic to a non-simply connected space; one has a trivial fundamental group and the other does not.

The Hawaiian earring can also be used to construct a semi-locally simply connected space that is not locally simply connected.

In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets.

Every simply connected space and every locally simply connected space is semilocally simply connected.

The results above for simply connected spaces can easily be extended to nilpotent spaces (whose fundamental group is nilpotent and acts nilpotently on the higher homotopy groups).

Both locally simply connected spaces and simply connected spaces are semi-locally simply connected, but neither converse holds.

If a space X is not simply connected, one can often rectify this defect by using its universal cover, a simply connected space which maps to X in a particularly nice way.

Another example of a non-semi-locally simply connected space is the complement of Q x Q in the Euclidean plane R, where Q denotes the set of rational numbers.

If tighter bounds on the sectional curvature are known, then this property generalizes to give a comparison theorem between geodesic triangles in M and those in a suitable simply connected space form; see Toponogov's theorem.

Compact toroids, especially the 'Field-Reversed Configuration' and the spheromak, attempt to combine the advantages of toroidal magnetic surfaces with those of a Simply connected space (non-toroidal) machine, resulting in a mechanically simpler and smaller confinement area.

The projective n-space is compact connected and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the n-sphere, a simply connected space.

Mathematicians knew that this property was unique to the 2-sphere, in the sense that any other simply connected space that does not have edges and is small enough (in mathematician terms, that is compact) is in fact the 2-sphere.

However, the two spaces cannot be homeomorphic because deleting a point from R leaves a non-simply connected space but deleting a point from R leaves a simply connected space (If we delete a line lying in R, the space wouldn't be simply connected anymore.