Circle of a sphere the Equator the only great circle. By contrast, all meridians of longitude, paired with their opposite meridian in the other hemisphere, form great circles. The diameter of the sphere which passes through the center of the circle is called its axis and the endpoints of this diameter are called its poles. A circle of a sphere can also be defined as the set of points at a given angular distance from a given pole. When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:
Sphere -ball; thus, it is homeomorphic to the Euclidean -sphere, but perhaps lacking its metric. The -sphere is denoted . It is an example of a compact topological manifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere. The Heine–Borel theorem implies that a Euclidean -sphere is compact. The sphere is the inverse image of a one-point set under the continuous function . Therefore, the sphere is closed. is also bounded; therefore it is compact. Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional