Thursday, January 7, 2010

Weak Convergence of Nets and Filters

Weak Convergence of Nets and Filters


 
Fawaz Ibrahim Abd-Hamid Hjouj




Basically, nets and filters are two substitutions for sequences, that overcome some of their undesirable features. The two general objects were used to characterize topological properties. Many researchers tried to facilitate the concept of convergence of nets and filters by introducing different types of convergence such as, weak convergence, rcconvergence, r-convergence, and others.
In this thesis, restriction is made to study weak convergence of nets and filters. The weak convergence structure of nets and filters is introduced. As well as their weak cluster points. Weak convergence need not imply the usual convergence, explicit examples of this approach are constructed. A number of results, whose statements are parallel to those in the usual sense of convergence, are established.
Some interesting results in regular spaces and Urysohn spaces have been achieved. Among of which, it is shown that in a regular space, weak convergence of nets is equivalent to convergence of nets. It is also proved that weak limits of nets or filters are unique in Urysohn spaces. Various mappings; strong continuous, closure continuous, weak continuous, and others; are all characterized by our present structure. As well, the connections between these maps and weak cluster points of nets and filters are investigated. Among of which is that: the closure continuous function between the spaces X and Y carries the weak cluster point of the net (x),) in X, into a weak cluster point off (xk) in Y.
It is a known fact that nets and filters are equivalent tools in the sense that filters can be used to construct nets and vice versa. In fact, there is a bridge between the two notions. The similarities between weak convergence of nets and filters -as well as their weak cluster points- are studied in details and are used in different places throughout the thesis.
Since the concept of convergence is not an intrinsic property of nets or filters, the weak convergence under various topologies is studied. Among of which, the weak convergence of nets and filters in the finite product topology. Many other weak versions of some topological properties that can be characterized by nets or filters are introduced in the obvious way. Among of which, the weak accumulation point of a subset. It is shown that: if E is a subset of a topological space X , then x E X is a weak accumulation point of E iff there is a net in (E - {x}) which converges weakly to x .
In addition, a weak version of almost open map is introduced under the name (weak closure open). It is shown that, if f: X-->Y is a map, where X is any space and Y is regular, then f is almost open iff f is weak closure open. A strong version of the strong closed graph concept, is defined under the name (supper strong closed graph). This concept is characterized by nets.
A weak version of compactness is introduced, without any axiom of separation to be assumed. This version of compactness is illustrated by explicit examples, and is characterized through nets and filters. It is also shown that, if X is a regular space, then X is compact iff it is weak compact. Moreover, a weak compact subset of a Urysohn space is closed. Finally, a weak version of compact preserving function is introduced under the name (compact weak preserving). Among the results in this approach is that: a weak continuous function between two topological spaces is compact weak preserving It is also proved that: the finite product of weak compact spaces is weak compact iff each factor of the product is weak compact.


Fawaz Ibrahim Abd-Hamid Hjouj
Supervisor
Dr. Bassam A. Manasrah
1999

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