The Topology Of The System Separation, Compactness And Convergence Structure

A new kind of T₂ separation axiom called strong T₂ separation in topological systems is introduced. Characterizations of T₂ separation by net convergences and by filter convergences are given. It is proved that strong T₂ separation and T₂ separation are all nice generalizations of classical T₂ separation for T₀ spaces. It is also proved that a Locale is a T₂ Locale iff it is a strong T₂ topological system. A subtle example is constructed to show that a Localic T₂ topological system may not be a strong T₂ topological system. It is obtained that spatializations, subsystems, topological sums and topological products of (strong) T₂ topological systems are still (strong) T₂ topological systems. Two kinds of compactness of topological systems—spatial compactness and localic compactness are also discussed. Characterizations and relations of the two compactnesses are given. It is proved that spatial compactness and localic compactness are all nice generalizations of compactness for topological spaces. It is also proved that a compact, T₂ topological system is a (strong) T3 and also a (strong) T₄ topological system. It is obtained that a closed subsystem of a compact topological system is a compact systemt, finite topological sum and topological products of compact topological systems are still compact topological systems. Finally, the concepts of the S?limits and the lower-limits on dcpos are generalized to the realm of posets. And a new concept ofφ-limits is introduced for posets. In terms of S-limits, the Scott topology, the way-below relation and the continuity of posets are characterized. The Lawson topology and the continuity of posets are also characterized in terms ofφ-limits. It is proved that a poset is continuous iff the S-limits for P areσ(P) topological determined. A meet continuous poset is continuous iff theφ-limits for P areλ(P) topological determined.