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Why do the cardinal functions exist in topological history? In other words, Why are they useful?

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Because they help us analyze cardinal numbers in set theory, and in topology it helps describe certain properties conveniently. –  MITjanitor Mar 18 '13 at 7:28
    
Googling for "topological history" gives this question, "Joyce and the Invention of Irish History: Finnegans Wake in Context", and "Social Cognition and Social Development: A Sociocultural Perspective" as the top three results, so it may not be the term you want... –  Trevor Wilson Mar 18 '13 at 7:30
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up vote 8 down vote accepted

The following is taken from R. Hodel's Cardinal Functions I in the Handbook of Set-Theoretic Topology (Kunen, Vaughan, eds; 1984), pp.1-61:

What are cardinal functions and why are they useful? Roughly speaking, cardinal functions extend such important topological properties as countable base, separable, and first countable to higher cardinality. Cardinal functions then allow one to formulate, generalize, and prove results of the type just discussed [every second-countable Hausdorff space has cardinality $\leq 2^\omega$; every first-countable separable Hausdorff space has cardinality $\leq 2^\omega$; every separable Hausdorff space has cardinality $\leq 2^{2^\omega}$] in a systematic and elegant manner. In addition, cardinal functions then allow one to make precise quantitative comparisons between certain topological properties. For example, it is well known that a space with a countable base has a countable dense set. A 'converse' of this result from the theory of cardinal functions states that a regular space with a countable dense set has a base of cardinality $\leq 2^\omega$. In summary, experience indicates that the idea of a cardinal function is one of the most useful and important unifying concepts in all of set-theoretic topology.


Addendum: As an example of this generalization in practice, recall the following familiar theorem about metrizable spaces.

Theorem. Given any metrizable space $X$, the following are equivalent:

  1. $X$ is second-countable.
  2. $X$ is separable.
  3. $X$ is Lindelöf.
  4. $X$ has no uncountable families of pairwise disjoint nonempty open sets.

Without too much difficulty one also achieves the following, more general, result.

Theorem. Given a metrizable space $X$, the following values are equal:

  1. $w(X)$, the weight of $X$.
  2. $d(X)$, the density of $X$.
  3. $L(X)$, the Lindelöf degree of $X$.
  4. $c(X)$, the cellularity of $X$.
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