Bi-quotient maps, countably bi-sequential spaces and related topics

Spaces Defined by Sequences

“…The advantage of the scheme lies in the uniformity of the definitions, giving rise to a general construction of pre-images that proves many of the classification theorems of [1]. The same construction also provides a way of classifying the spaces introduced in [6], as well as the quotient images of paracompact A/-spaces, A/-spaces and <?-spaces, without any separation axioms.…”

“…6-7]. If the present definition is not the same as in [6], the equivalence is either proved in [1] or is easy to show. Definition 2.6.…”

“…In [1], Michael discussed various generalizations of first countable spaces and gave an extensive collection of theorems providing some answers to the question: "What kind of spaces are the images of various familiar classes of spaces under certain types of quotient maps?" Olson [6] continued this investigation and extended it to some similarly defined classes of spaces which generalize the <7-spaces introduced in [2]. Most of the spaces involved are defined by conditions requiring the existence of certain types of nested, or decreasing, sequences of sets.…”

“…As a corollary (again using theorems of [1]) we get a further relationship between the spaces introduced for these classifications and /:-spaces. §6 raises a question con-cerning regular spaces suggested by these classifications and by a question from [6].…”

“…Most of the implications among these classes of spaces have been reported in [1] and [6]. Among the new classes defined, it is obvious that X is sequential => X is sequentially-/: =^> X is sequentially quasi-/: => X is sequentially-c7.…”