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In geometry, a disk (also spelled disc) [1] is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not. [2] For a radius, , an open disk is usually denoted as and a closed disk is ¯.
The spherical isoperimetric inequality states that (), and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.
This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed.
An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it contains neither. The interval [0, 1) = {x | 0 ≤ x < 1}, for example, is left-closed and right-open. The empty set and the set of all reals are both open and ...
A closed rectangle V is not a neighbourhood of any of its corners or its boundary since there is no open set in V containing any corner or edge point. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.
The closed disk is a simple example of a surface with boundary. The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface.
A closed half-space is a set in the form {()} or {()}, and likewise an open half-space uses strict inequality. [11] [12] Half-spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex cone C that is not the whole space V must be contained in some closed half-space H of V; this is a special case of Farkas' lemma.
Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is: A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.