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In topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, a branch of mathematics, the infinite broom is a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
that is used as an example distinguishing various notions of connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be ...
. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space.Chapter 6 exercise 3.5 of
Definition
The infinite broom is the subset of the Euclidean plane that consists of allclosed line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between it ...
s joining the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
to the point as ''n'' varies over all positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, 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 or to the point .
Properties
Both the infinite broom and its closure are Connected space, connected, as everyopen set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
in the plane which contains the segment on the ''x''-axis must intersect slanted segments. Neither are locally connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedne ...
. Despite the closed infinite broom being arc connected
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 non-empty open subsets. Connectedness is one of the principal topological properties ...
, the standard infinite broom is not path connected
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 non-empty open subsets. Connectedness is one of the principal topological properties ...
.
The interval ,1on the ''x''-axis is a deformation retract
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deform ...
of the closed infinite broom, but it is not a ''strong'' deformation retract.
See also
*Comb space
In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb spa ...
* Integer broom topology In general topology, a branch of mathematics, the integer broom topology is an example of a topology on the so-called integer broom space ''X''.
Definition of the integer broom space
The integer broom space ''X'' is a subset of the plane R ...
* List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that ...
References
{{Reflist Topological spaces Trees (topology)