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The integer broom space ''X'' is a
general topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
, 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 integer broom topology is an example of a 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 ...
on the so-called integer broom space ''X''.
Definition of the integer broom space
The integer broom space ''X'' 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 plane R2. Assume that the plane is parametrised by polar coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are
*the point's distance from a reference ...
. The integer broom contains the origin and the points such that ''n'' is a non-negative integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
and , where Z+ is the set of positive integers. The image on the right gives an illustration for and . Geometrically, the space consists of a collection of convergent sequence
As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n \times \sin\left(\tfrac1\right) equals 1."
In mathematics, the li ...
s. For a fixed ''n'', we have a sequence of points − lying on circle with centre (0, 0) and radius ''n'' − that converges to the point (''n'', 0).
Definition of the integer broom topology
We define the topology on ''X'' by means of aproduct topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. The integer broom space is given by the polar coordinates
:
Let us write for simplicity. The integer broom topology on ''X'' is the product topology induced by giving ''U'' the right order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, t ...
, and ''V'' the subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
from R.
Properties
The integer broom space, together with the integer broom topology, is acompact topological space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
. It is a T0 space, but it is neither a T1 space nor a Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
. The space is 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 ...
, while neither 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 ...
nor 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 ...
.
See also
* Comb space *Infinite broom
In topology, a branch of mathematics, 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 re ...
* 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
{{DEFAULTSORT:Integer Broom topology General topology Topological spaces