The Jordan curve theorem fails for a simply connected one of these objects named for Alexander.
As a cell complex, this object is comprised of one 0-cell and one n-cell, and is formed by quotienting by the boundary of the n-cell. The quotient map from this object to SU(2) / U(1) ["S-U-two mod U-one"] is the Hopf fibration. The Jordan curve theorem fails for a simply connected one of these objects named for Alexander. Stephen Smale proved that there exists immersions of this object that are regularly homotopic in achieving its (*) eversion. As a smooth manifold, the coordinate maps for this object are given by stereographic projections, and the complex projective line forms one of these objects named for Riemann. Every closed, simply connected three dimensional manifold is homeomorphic to one of these objects according to the Poincare conjecture. For ten points, name this object, the locus of all points equidistant from a center.
As a cell complex, this object is comprised of one 0-cell and one n-cell, and is formed by quotienting by the boundary of the n-cell.
Sphere [or Horned Sphere; or 3-sphere; or n-sphere or Sn (S-n), where n is any positive whole number]