Infinite broom
Encyclopedia
In topology
, the infinite broom is a subset
of the Euclidean plane that is used as an example distinguishing various notions of connectedness
. The closed infinite broom is the closure
of the infinite broom, and is also referred to as the broom space.
to the point (1, 1 / n) as n varies over all positive integers, together with the interval
(½, 1] on the x-axis.
The closed infinite broom is then the infinite broom together with the interval (0, ½] on the x-axis. In other words, it consists of all closed line segments joining the origin to the point (1, 1 / n) or to the point (1, 0).
, as every open set in the plane which contains the segment on the x-axis must intersect slanted segments. Neither are locally connected. Despite the closed infinite broom being arc connected, the standard infinite broom is not path connected.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the infinite broom is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of the Euclidean plane that is used as an example distinguishing various notions of connectedness
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. The closed infinite broom is the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of the infinite broom, and is also referred to as the broom space.
Definition
The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the originOrigin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
to the point (1, 1 / n) as n varies over all positive integers, together with the interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
(½, 1] on the x-axis.
The closed infinite broom is then the infinite broom together with the interval (0, ½] on the x-axis. In other words, it consists of all closed line segments joining the origin to the point (1, 1 / n) or to the point (1, 0).
Properties
Both the infinite broom and its closure are connectedConnected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, as every open set in the plane which contains the segment on the x-axis must intersect slanted segments. Neither are locally connected. Despite the closed infinite broom being arc connected, the standard infinite broom is not path connected.