In mathematics, particularly
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 comb space is a particular
subspace of
that resembles a
comb
A comb is a tool consisting of a shaft that holds a row of teeth for pulling through the hair to clean, untangle, or style it. Combs have been used since Prehistory, prehistoric times, having been discovered in very refined forms from settlemen ...
. The comb space has properties that serve as a number of
counterexamples
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...
. The
topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functi ...
has similar properties to the comb space. The deleted comb space is a variation on the comb space.
Formal definition
Consider
with its
standard topology
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vecto ...
and let ''K'' be the
set . The set ''C'' defined by:
:
considered as a subspace of
equipped with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
is known as the comb space. The deleted comb space, D, is defined by:
:
.
This is the comb space with the line segment