View source for Bifactorial mapping

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161801.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161802.png" /> into a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161803.png" />, in which for any covering of the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161804.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161805.png" /> by sets open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161806.png" /> it is possible to select a finite number of sets so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161807.png" /> is located inside the image of their union. It is particularly important that the product of any collection of bifactorial mappings is a bifactorial mapping. Bifactorial mappings constitute an extensive class of factorial mappings, but nevertheless preserve the fine topological properties of spaces. Thus, continuous bifactorial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161808.png" />-mappings preserve a pointwise-countable base, and a factorial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016180/b0161809.png" />-mapping of a space with a pointwise-countable base onto a space of pointwise-countable type is bifactorial.