On π-Images of Locally Separable Metric Spaces

To determine what spaces are the images of “nice” spaces under “nice” mappings is one of the central questions of general topology in 1 . In the past, many noteworthy results on images of metric spaces have been obtained. For a survey in this field, see 2 , for example. A characterization for a quotient compact image of a locally separable metric space is obtained in 3 . Also, such a quotient image is precisely a pseudo-sequence-covering quotient compact image of a locally separable metric space 4 . Recently, π-images of metric spaces cause attention once again in 5, 6 . It is known that a space is a compact-covering resp., sequence-covering, pseudo-sequence-covering, sequentially-quotient π-image of a metric space if and only if it has a point-star network consisting of cfp-covers resp., cs-covers, wcs-covers, cs∗-covers 5– 7 . In a personal communication, the first author of 6 informs that it seems to be difficult to obtain “nice” characterizations of “nice” images of locally separable metric spaces instead of metric or locally compact metric domains . Thus, we are interested in the following question.