Pretopological space explained
In general topology, a pretopological space is a generalization of the concept of topological space. 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, and is covered in the article on that topic.
Let
be a set. A
neighborhood system for a
pretopology on
is a collection of
filters
one for each element
of
such that every set in
contains
as a member. Each element of
is called a
neighborhood of
A pretopological space is then a set equipped with such a neighborhood system.
converges to a point
in
if
is eventually in every neighborhood of
A pretopological space can also be defined as
a set
with a preclosure operator (
Čech closure operator)
The two definitions can be shown to be equivalent as follows: define the closure of a set
in
to be the set of all points
such that some net that converges to
is eventually in
Then that closure operator can be shown to satisfy the axioms of a preclosure operator. Conversely, let a set
be a neighborhood of
if
is not in the closure of the complement of
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.
A map
f:(X,\operatorname{cl})\to(Y,\operatorname{cl}')
between two pretopological spaces is
continuous if it satisfies for all subsets
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.
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