In the 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 ...
known as
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 ...
, the topologist's sine curve or Warsaw sine curve is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
with several interesting properties that make it an important textbook example.
It can be defined as the
graph of the function
on the
half-open interval , together with the origin, under the topology
subspace topology, induced from 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 ...
:
:
Properties
The topologist's sine curve is
connected but neither
locally connected nor
path connected. This is because it includes the point but there is no way to link the function to the origin so as to make a
path.
The space is the continuous image of a
locally compact space (namely, let be the space
and use the map
defined by
and
for ), but is not locally compact itself.
The
topological dimension of is 1.
Variants
Two variants of the topologist's sine curve have other interesting properties:
*The ''closed'' topologist's sine curve can be defined by taking the topologist's sine curve and adding its set of
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
s,
; some texts define the topologist's sine curve itself as this closed version, as they prefer to use the term 'closed topologist's sine curve' to refer to another curve.
This space is closed and bounded and so
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
by the
Heine–Borel theorem, but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.
*The ''extended'' topologist's sine curve can be defined by taking the closed topologist's sine curve and adding to it the set
. This variant is
arc connected but not
locally connected.
See also
*
List of topologies
*
Warsaw circle
References
*
*{{mathworld, urlname=TopologistsSineCurve, title=Topologist's Sine Curve
Topological spaces