In the
mathematical
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 ...
field of
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 ...
, 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 ...
is usually defined by declaring its
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of
closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in
Kazimierz Kuratowski
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Biography and studies
Kazimierz Kuratowski was born in Warsaw, (t ...
's well-known textbook on
point-set topology
In mathematics, general 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 differential topology, geomet ...
, a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom. Likewise, the neighborhood-based axioms (in the context of
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s) can be retraced to
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
's original definition of a topological space in
Grundzüge der Mengenlehre.
Many different textbooks use many different inter-dependences of concepts to develop point-set topology. The result is always the same collection of objects: open sets, closed sets, and so on. For many practical purposes, the question of which foundation is chosen is irrelevant, as long as the meaning and interrelation between objects (many of which are given in this article), which are the same regardless of choice of development, are understood. However, there are cases where it can be useful to have flexibility. For instance, there are various natural notions of
convergence of measures
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...
, and it is not immediately clear whether they arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence.
Standard definitions via open sets
A topological space is a set
together with a collection
of
subsets of
satisfying:
* The
empty set and
are in
* The
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of any collection of sets in
is also in
* The
intersection of any pair of sets in
is also in
Equivalently, the intersection of any finite collection of sets in
is also in
Given a topological space
one refers to the elements of
as the open sets of
and it is common to only refer to
in this way, or by the label topology. Then one makes the following secondary definitions:
* Given a second topological space
a function
is said to be continuous if and only if for every open subset
of
one has that
is an open subset of
* A subset
of
is closed if and only if its complement
is open.
* Given a subset
of
the closure is the set of all points such that any open set containing such a point must intersect
* Given a subset
of
the interior is the union of all open sets contained in
* Given an element
of
one says that a subset
is a neighborhood of
if and only if
is contained in an open subset of
which is also a subset of
Some textbooks use "neighborhood of
" to instead refer to an open set containing
* One says that a net converges to a point
of
if for any open set
containing
the net is eventually contained in
* Given a set
a
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
is a collection of nonempty subsets of
that is closed under finite intersection and under supersets. Some textbooks allow a filter to contain the empty set, and reserve the name "proper filter" for the case in which it is excluded. A topology on
defines a notion of a filter converging to a point
of
by requiring that any open set
containing
is an element of the filter.
* Given a set
a filterbase is a collection of nonempty subsets such that every two subsets intersect nontrivially and contain a third subset in the intersection. Given a topology on
one says that a filterbase converges to a point
if every neighborhood of
contains some element of the filterbase.
Definition via closed sets
Let
be a topological space. According to
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
, the collection
of closed sets satisfies the following properties:
* The
empty set and
are elements of
* The
intersection of any collection of sets in
is also in
* The
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of any pair of sets in
is also in
Now suppose that
is only a set. Given any collection
of subsets of
which satisfy the above axioms, the corresponding set
is a topology on
and it is the only topology on
for which
is the corresponding collection of closed sets. This is to say that a topology can be defined by declaring the closed sets. As such, one can rephrase all definitions to be in terms of closed sets:
* Given a second topological space
a function
is continuous if and only if for every closed subset
of
the set
is closed as a subset of
* a subset
of
is open if and only if its complement
is closed.
* given a subset
of
the closure is the intersection of all closed sets containing
* given a subset
of
the interior is the complement of the intersection of all closed sets containing
Definition via closure operators
Given a topological space
the closure can be considered as a map
where
denotes the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
One has the following
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first forma ...
:
*
*
*
*
If
is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of cl satisfies the previous axioms for closed sets, and hence defines a topology; it is the unique topology whose associated closure operator coincides with the given cl. As before, it follows that on a topological space
all definitions can be phrased in terms of the closure operator:
*Given a second topological space
a function
is continuous if and only if for every subset
of
one has that the set
is a subset of
* A subset
of
is open if and only if
* A subset
of
is closed if and only if
* Given a subset
of
the interior is the complement of
Definition via interior operators
Given a topological space
the interior can be considered as a map
where
denotes the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
It satisfies the following conditions:
*
*
*
*
If
is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of int satisfies the previous axioms for open sets, and hence defines a topology; it is the unique topology whose associated interior operator coincides with the given int. It follows that on a topological space
all definitions can be phrased in terms of the interior operator, for instance:
* Given topological spaces
and
a function
is continuous if and only if for every subset
of
one has that the set
is a subset of
* A set is open if and only if it equals its interior.
* The closure of a set is the complement of the interior of its complement.
Definition via neighbourhoods
Recall that this article follows the convention that a
neighborhood is not necessarily open. In a topological space, one has the following facts:
* If
is a neighborhood of
then
is an element of
* The intersection of two neighborhoods of
is a neighborhood of
Equivalently, the intersection of finitely many neighborhoods of
is a neighborhood of
* If
contains a neighborhood of
then
is a neighborhood of
* If
is a neighborhood of
then there exists a neighborhood
of
such that
is a neighborhood of each point of
.
If
is a set and one declares a nonempty collection of neighborhoods for every point of
satisfying the above conditions, then a topology is defined by declaring a set to be open if and only if it is a neighborhood of each of its points; it is the unique topology whose associated system of neighborhoods is as given. It follows that on a topological space
all definitions can be phrased in terms of neighborhoods:
* Given another topological space
a map
is continuous if and only for every element
of
and every neighborhood
of
the preimage
is a neighborhood of
* A subset of
is open if and only if it is a neighborhood of each of its points.
* Given a subset
of
the interior is the collection of all elements
of
such that
is a neighbourhood of
.
* Given a subset
of
the closure is the collection of all elements
of
such that every neighborhood of
intersects
Definition via convergence of nets
Convergence of nets satisfies the following properties:
# Every constant net converges to itself.
# Every
subnet
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting.
Computers that belong to the same subnet are addressed with an identical ...
of a convergent net converges to the same limits.
# If a net does not converge to a point
then there is a subnet such that no further subnet converges to
Equivalently, if
is a net such that every one of its subnets has a sub-subnet that converges to a point
then
converges to
# '/''Convergence of iterated limits''. If
in
and for every
is a net that
converges to in
then there exists a diagonal net that converges to
(a ''diagonal net'' is a subnet of
where the domain of this net is
ordered lexicographically first by
and then by
explicitly, given
declare that
holds if and only if both (1)
and also (2) if
then
).
If
is a set, then given any collection of nets and points satisfying the above axioms, a closure operator on
is defined by sending any given set
to the set of all limits of all nets in
the corresponding topology is the unique topology inducing the given convergences of nets to points.
Given a subset
of a topological space
*
is open in
if and only if every net converging to an element of
is eventually contained in
* the closure of
in
is the set of all limits of all convergent nets in
*
is closed in
if and only if there does not exist a net in
that converges to an element of the complement
A subset
is closed in
if and only if every limit point of every convergent net in
necessarily belongs to
A function
between two topological spaces is continuous if and only if for every
and every net
in
that
converges to in
the net
converges to
in
Definition via convergence of filters
A topology can also be defined on a set by declaring which filters converge to which points. One has the following characterizations of standard objects in terms of
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
and
prefilter
In mathematics, a filter on a set X is a family \mathcal of subsets such that:
# X \in \mathcal and \emptyset \notin \mathcal
# if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal
# If A,B\subset X,A\in \mathcal, and A\subset B, then ...
s (also known as filterbases):
* Given a second topological space
a function
is continuous if and only if it preserves
convergence of prefilters.
* A subset
of
is open if and only if every
filter converging to an element of
contains
* A subset
of
is closed if and only if there does not exist a prefilter on
which converges to a point in the complement
* Given a subset
of
the closure consists of all points
for which there is a prefilter on
converging to
* A subset
of
is a neighborhood of
if and only if it is an element of every filter converging to
See also
*
*
*
*
*
References
Notes
Textbooks
*
*
*
*
*
General topology
Categories in category theory