page_title: Manifold
section_title: Circle
hierarchical_section_title: Manifold / Motivating examples / Circle
context_page_description: In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.
  One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
  Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may be not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane.
context_section_description: After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x² + y² = 1, where the y-coordinate is positive (indicated by the yellow circular arc in Figure 1). Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the upper arc to the open interval (−1, 1):
  Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle:
  Together, these parts cover the whole circle and the four charts form an atlas for the circle.
  The top and right charts,  and  respectively, overlap in their domain: their intersection lies in the quarter of the circle where both the - and the -coordinates are positive. Each map this part into the interval , though differently. Thus a function  can be constructed, which takes values from the co-domain of  back to the circle using the inverse, followed by the  back to the interval.  Let a be any number in , then:
  Such a function is called a transition map.
  The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts
  and
  Here s is the slope of the line through the point at coordinates (x, y) and the fixed pivot point (−1, 0); similarly, t is the opposite of the slope of the line through the points at coordinates (x, y) and (+1, 0). The inverse mapping from s to (x, y) is given by
  It can easily be confirmed that x² + y² = 1 for all values of s and t. These two charts provide a second atlas for the circle, with the transition map
  (that is, one has this relation between s and t for every point where s and t are both nonzero).
  Each chart omits a single point, either (−1, 0) for s or (+1, 0) for t, so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.
caption: Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.