general topology In mathematics, general topology (or point set 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 differ ...

, a pretopological space is a generalization of the concept of

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 ...

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

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

X

be a set. A

neighborhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...

for a pretopology on

X

is a collection of

filter Filtration is a physical process that separates solid matter and fluid from a mixture. Filter, filtering, filters or filtration may also refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Fil ...

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 may refer to: Broadcast media United States * National Educational Television, the predecessor of the Public Broadcasting Service (PBS) in the United States * National Empowerment Television, a politically conservative cable TV network ...

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 Čech (feminine Čechová) is a Czech surname meaning Czech. It was used to distinguish an inhabitant of Bohemia from Slovaks, Moravians and other ethnic groups. Notable people with the surname include: * Dana Čechová (born 1983), Czech table t ...

)

\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