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In topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a branch of mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the infinite broom is a subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
of the Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
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 endpoints (its extreme points), and contains every point on the line that is between its endpoints. It is a special case of an '' arc'', with zero curvat ...
s joining the origin to the point as ''n'' varies over all positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
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, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
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 of open connected sets.
As a stronger notion, the space ''X'' is locally path connected if ev ...
. Despite the closed infinite broom being arc connected
Arc may refer to:
Mathematics
* Arc (geometry), a segment of a differentiable curve
** Circular arc, a segment of a circle
* Arc (topology), a segment of a path
* Arc length, the distance between two points along a section of a curve
* Arc (pr ...
, 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 (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
.
The interval ,1on the ''x''-axis is a deformation retract
In topology, 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 deformation retraction is a mappi ...
of the closed infinite broom, but it is not a ''strong'' deformation retract.
See also
* Comb space * Integer broom topology *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)