The following is a list of named
topologies or
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 ...
s, many of which are counterexamples 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 ...
and related branches 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 ...
. This is not a list of
properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Philosophy and science
* Property (philosophy), in philosophy and logic, an abstraction characterizing an ...
that a topology or topological space might possess; for that, see
List of general topology topics and
Topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
.
Discrete and indiscrete
*
Discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
− All subsets are open.
*
Indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, chaotic topology, or
Trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
− Only the empty set and its complement are open.
Cardinality and ordinals
*
Cocountable topology
The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set X. In this topology, a set is open if its complement in X is either countable or equal to the entire set. Equivalen ...
** Given a topological space
the ''
'' on
is the topology having as a subbasis the union of and the family of all subsets of
whose complements in
are countable.
*
Cofinite topology
*
Double-pointed cofinite topology
*
Ordinal number topology
*
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. ...
*
Ran space
*
Tychonoff plank
Finite spaces
*
Discrete two-point space − The simplest example of a
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
.
*
Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are ...
*
Pseudocircle − A
finite topological space
In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are ...
on 4 elements that fails to satisfy any
separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...
besides
T0. However, from the viewpoint of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, it has the remarkable property that it is indistinguishable from the
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
*
Sierpiński space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.
The ...
, also called the
connected two-point set − A 2-point set
with the
particular point topology
Integers
*
Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e.
) for which there is no sequence in
that converges to
but there is a sequence
in
such that
is a cluster point of
*
Arithmetic progression topologies
*
The Baire space −
with the product topology, where
denotes the
natural numbers
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 ...
endowed with the discrete topology. It is the space of all sequences of natural numbers.
*
Divisor topology
*
Partition topology
In mathematics, a partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names:
* The is t ...
**
Deleted integer topology
**
Odd–even topology
Fractals and Cantor set
*
Apollonian gasket
*
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883.
Throu ...
− A subset of the closed interval