Abstract
A function φ which assigns a cardinal φ (X) to each topological space X is called a cardinal function or a cardinal invariant if it is a topological. The values of cardinal functions are assumed to always be infinite cardinals. The simplicity of infinite cardinal arithmetic helps to simplify things. This chapter adopts the set-theoretic notations that cardinal numbers are initial ordinals, that is, κ is a cardinal if and only if (iff) it is the smallest ordinal of the cardinality |κ|. Symbols like κ,λ, always denote cardinal numbers. ω and ω1 are used to denote the first infinite ordinal (cardinal) and the first uncountable ordinal (cardinal) respectively. Therefore, for each cardinal κ, there is κ + ω =κ if κ is infinite, and κ + ω =ω if κ is finite. Global and local cardinal functions are defined using this relation. For compact spaces, some cardinal functions can be characterized by the existence of a family with some weaker condition.
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