In
general 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 polytopological space consists of a
set together with a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of
topologies on
that is
linearly ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
by the
inclusion relation
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
(
is an arbitrary
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consis ...
). It is usually assumed that the topologies are in non-decreasing order,
but some authors prefer to put the associated
closure operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
in non-decreasing order (operators
and
satisfy
if and only if
for all
),
in which case the topologies have to be non-increasing.
Polytopological spaces were introduced in 2008 by the philosopher
Thomas Icard
Thomas may refer to:
People
* List of people with given name Thomas
* Thomas (name)
* Thomas (surname)
* Saint Thomas (disambiguation)
* Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church
* Thomas the Ap ...
for the purpose of defining a topological
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
of
Japaridze's polymodal logic (GLP).
They subsequently became an object of study in their own right, specifically in connection with
Kuratowski's closure-complement problem.
Definition
An
-topological space
is a set
together with a
monotone map Top
where
is a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
and Top
is the set of all possible topologies on
ordered by inclusion. When the partial order
is a linear order, then
is called a polytopological space. Taking
to be the
ordinal number an
-topological space can be thought of as a set
together with
topologies
on it (or
depending on preference). More generally, a multitopological space
is a set
together with an arbitrary family
of topologies on
See also
*
Bitopological space
References
{{reflist
Topology