In the branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
known as
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 ...
, the topologist's sine curve or Warsaw sine curve is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
with several interesting properties that make it an important textbook example.
It can be defined as the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of the function sin(1/''x'') on the
half-open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
(0, 1], together with the origin, under the topology
subspace topology, induced from the
Euclidean plane:
:
Properties
The topologist's sine curve ''T'' is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
but 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 entirely of open, connected sets.
Background
Throughout the history of topology, connectedness ...
nor
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 that ...
. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
.
The space ''T'' is the continuous image of a
locally compact space (namely, let ''V'' be the space ∪ (0, 1
], and use the map ''f'' from ''V'' to ''T'' defined by
''f''(−1) = (0,0) and
''f''(''x'') =
(''x'', sin(1/''x'')) for ''x'' > 0), but ''T'' is not locally compact itself.
The
topological dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
of ''T'' 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 with respect to the topology on X also contai ...
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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
. It is
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 t ...
but not
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, connectedness ...
.
See also
*
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, ...
*
Warsaw circle
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory ...
References
*
*{{mathworld, urlname=TopologistsSineCurve, title=Topologist's Sine Curve
Topological spaces