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 pretopological space is a generalization of the concept of

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 ...

. A pretopological space can be defined in terms of either filters or a

preclosure operator In topology, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the f ...

. The similar, but more abstract, notion of a Grothendieck pretopology is used to form a

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is c ...

, and is covered in the article on that topic. Let

X

be a set. A neighborhood system for a pretopology on

X

is a collection of

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 ...

N(x),

one for each element

x

X

such that every set in

N(x)

contains

x

as a member. Each element of

N(x)

is called a neighborhood of

x.

A pretopological space is then a set equipped with such a neighborhood system. A

net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded u ...

x_

converges to a point

x

X

x_

is eventually in every neighborhood of

x.

A pretopological space can also be defined as

(X, \operatorname),

a set

X

with a preclosure operator (

Čech closure operator In topology, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the fou ...

)

\operatorname.

The two definitions can be shown to be equivalent as follows: define the closure of a set

S

X

to be the set of all points

x

such that some net that converges to

x

is eventually in

S.

Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set

S

be a neighborhood of

x

x

is not in the closure of the complement of

S.

The set of all such neighborhoods can be shown to be a neighborhood system for a pretopology. A pretopological space is a topological space when its closure operator is

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

. A map

f : (X, \operatorname) \to (Y, \operatorname')

between two pretopological spaces is continuous if it satisfies for all subsets

A \subseteq X,

f(\operatorname(A)) \subseteq \operatorname'(f(A)).

References

* E. Čech, ''Topological Spaces'', John Wiley and Sons, 1966. * D. Dikranjan and W. Tholen, ''Categorical Structure of Closure Operators'', Kluwer Academic Publishers, 1995. * S. MacLane, I. Moerdijk, ''Sheaves in Geometry and Logic'', Springer Verlag, 1992.

External links

Recombination Spaces, Metrics, and Pretopologies
B.M.R. Stadler, P.F. Stadler, M. Shpak., and G.P. Wagner. (See in particular Appendix A.)
Closed sets and closures in Pretopology
M. Dalud-Vincent, M. Brissaud, and M Lamure. 2009 . {{Topology General topology

See also

References

External links