Pretopological space
Encyclopedia
In general topology
, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined as in terms of either filters or a preclosure operator
.
Let X be a set. A neighborhood system for a pretopology on X is a collection of filter
s N(x), one for each element of 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
xα converges to a point x in X if xα is eventually in every neighborhood of x.
A pretopological space can also be defined as (X, cl ), a set X with a preclosure operator (Čech closure operator) cl. The two definitions can be shown to be equivalent as follows: define the closure of a set S in 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 if 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.
A map f : (X, cl ) → (Y, cl ) between two pretopological spaces is continuous if it satisfies for all subsets A of X:
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined as in terms of either filters or a preclosure operator
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...
.
Let X be a set. A neighborhood system for a pretopology on X is a collection of filter
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
s N(x), one for each element of 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 (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
xα converges to a point x in X if xα is eventually in every neighborhood of x.
A pretopological space can also be defined as (X, cl ), a set X with a preclosure operator (Čech closure operator) cl. The two definitions can be shown to be equivalent as follows: define the closure of a set S in 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 if 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.
A map f : (X, cl ) → (Y, cl ) between two pretopological spaces is continuous if it satisfies for all subsets A of X:
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.)