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Square brackets are also often used in place of a second set of parentheses when they are nested—so as to provide a visual distinction. In mathematical expressions in general, parentheses are also used to indicate grouping (i.e., which parts belong together) when edible to avoid ambiguities and improve clarity.
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 ¯.
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 ...
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.
[45] [46] The endpoint adjoining the square bracket is known as closed, whereas the endpoint adjoining the parenthesis is known as open. [ 44 ] In group theory and ring theory , brackets denote the commutator .
A chart of a manifold is a homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the unit sphere in R 3 {\displaystyle \mathbb {R} ^{3}} with a single point removed and the set of all points in R 2 {\displaystyle \mathbb {R} ^{2}} (a ...
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A non-closed curve may also be called an open curve. If the domain of a topological curve is a closed and bounded interval = [,], the curve is called a path, also known as topological arc (or just arc). A curve is simple if it is the image of an interval or a circle by an injective continuous function.