Sub-Separation Axioms In Pre-topological Spaces And Strong-T₁Compactifications Of Pretopological Spaces

The aim of this paper is to study sub-separation axioms in pre-topological spaces. These axioms are weaker than the corresponding separation axioms in topological spaces. Sub-separation axioms, such as sub-T2, sub-regular, sub-normal, and hereditarily sub-normal, are defined in pre-topological spaces, re-lations between them are discussed in detail, hereditary property and multiplicative property of these sub-separation axioms and connections between them and strong-T1compactifications of pre-topological spaces and nonstandard compactifications of topological spaces are studied.The key points and the main contents of this paper are as follows:In the first chapter, we give the basic concepts and results of the theory of pre-topological spaces and Wallman compactifications, and nonstandard compacti-fications which are used in the whole paper.In the second chapter, firstly, sub-separation axioms sub-T2, sub-regular, sub-normal, and hereditarily sub-normal are defined in pre-topological spaces by using the way of topology, and the relationship between them are deeply discussed. Then the properties of hereditary, quotient and multiplicative are discussed.In the third chapter, the definition of strong-T1is given first in pre-topological spaces, then strong-T1compactifications of pre-topological spaces are dicussed. Fi-nally, nonstandard compactifications of these Sub-separation axioms in topological spaces are studied.