Perfect mappings and certain interior images of 𝑀-spaces

Worrell, J. M.; Wicke, H. H.

doi:10.1090/S0002-9947-1973-0320990-3

Perfect mappings and certain interior images of $M$-spaces
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Trans. Amer. Math. Soc. 181 (1973), 23-35 Request permission

Abstract:

The main theorems of this paper show that certain conditions (called ${\lambda _c},{\lambda _b},{\beta _c}$, and ${\beta _b}$) are invariant, in the presence of ${T_0}$-regularity, under the application of closed continuous peripherally compact mappings. Interest in these conditions lies in the fact that they may be used to characterize certain regular ${T_0}$ open continuous images of some classes of $M$-spaces in the sense of K. Morita, and in the fact that they are preserved by open continuous mappings with certain appropriate additional conditions. For example, the authors have shown that a regular ${T_0}$-space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition ${\lambda _b}$ [Pacific J. Math. 37 (1971), 265-275]. Moreover, in the same paper it is shown that if a completely regular ${T_0}$-space satisfies condition ${\lambda _b}$ then any ${T_0}$ completely regular open continuous image of it also satisfies ${\lambda _b}$. These results together with the results of the present paper and certain known results lead to the following theorem: The smallest subclass of the class of regular ${T_0}$-spaces which contains all paracompact Čech complete spaces and which is closed with respect both to the application of perfect mappings and to the application of open continuous mappings preserving ${T_0}$-regularity is the subclass satisfying condition ${\lambda _b}$. Similar results are obtained for the regular ${T_0}$-spaces satisfying ${\lambda _c},{\beta _b}$, and ${\beta _c}$. The other classes of $M$-spaces involved are the regular ${T_0}$ complete $M$-spaces (i.e., spaces which are quasi-perfect preimages of complete metric spaces), ${T_2}$ paracompact $M$-spaces, and regular ${T_0}M$-spaces. In the last two cases besides the inferiority of the mappings the notion of uniform $\lambda$-completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85-100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.

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A perfect mapping not preserving the $p$-space property