  A common generalization for two of the main streams of cardinality
inequalities is developed; each stream derives from the famous inequality
established by A.V. Arhangel'ski\u{\i} in 1969 for Hausdorff spaces. At the end
of one stream is the recent inequality by Bella and at the end of the second
stream is the 1988 inequality by Bella and Cammaroto. This generalization is
extended and used to analyze a result containing an increasing chain of spaces
that satisfies the same cardinality inequality. The paper is concluded with
some open problems.
