Inverse Limit Of Topological Spaces And Tychonoff Product Space

In 1990, a famous topologist K. Chiba proved that the topological property such as normality , collectionwise normal are invariable under the inverse limit operation. Since the last decade , the topologist Zhaohui Xiong and Jiguang Jiang on those topological spaces such as normalσ-collectionwise normal,δ-normal and strictly quasiparacom -pact etc., which were pedicted by some closed coverage, achieved a series of important results on the preserving of inverse limit. Thereupon, naturally next question become the hot questions investigatived by scholars.Question 1: Under what conditions, the expandable spaces can be preserving under the inverse limit operation?In 2003 and 2004, Peiyong Zhu proved that almost expandable and expandable space can be preserving under the inverse limit operation if the space isλ-paracompact.In 2001and 2004, he obtained the characterization of Tychonoff product on both normal strictly quasiparacompact space and expandable space. And then , we given birth to the following questions.Question 2: Compared with both expandable space and mesocompact space, whether the cf -expandable space class can keep the invariable property under inverse limit operation ?Question 3: Whether the property b1 space weaker than normal strictly quasi-paracompact space and strictly quasi-paracompact space , have the similar invariable property of inverse limit and good Tychonoff product property with those space ?Let X be the limit of inverse system { Xα,πβα,Λ}, if eachπαbe onto and open mapping andλ=Λ,we obtain the following results:(1) If X is (hereditarily)λ-paracompact and each Xαis(hereditarily)θ-cf expand -able space, then X is(hereditarily)θ-cf expandable space .(2) If X is (hereditarily)λ-paracompact and each Xαis (hereditarily)property b1 space, then X is(hereditarily)property b1 space .Secondly, on the bases of inverse limit result , we obtained a batch of equivalent characterization on correspondable topological space class.In this paper, a series of research results, to a certain extent, enriched and developed the topological space inverse limit theories and Tychonoff product theories. It is more complement and perfect of the general topology product theory .