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But since introducing new tags appeared to be so delicate question, then I agree to keep just wide tags: "set-theory" and "general-topology". This is less important comparing to the presence or absence of interesting questions (and answers) in these fields.
"cardinal-function-topology" is a separated topic covering cardinal faunctions on topological spaces like weight, separability, cellularity, etc. "cardinal characteristics of the continuum" is a bit different set-theoretic topic decribing combinatorial cardinality properties of some standard Polish spaces like real line, Cantor set, Baire space. Up to my understanding "cardinal characteristic of continuum" is a subtopic of "small-uncountable-cardinals". The latter relates to cardinals that are near the continuum.
Ok, I have no energy (and arguments) to discuss this further. Maybe you are right. Now what would you suggest to do? Remove all (I hope 8) tags and wait 24 hours till it will disappear? Unfortunately removing tags will bump the corresponding problems to the top, which makes some MO-users very nervous :(
Like "cardinal characteristics of the continuum" (the title of the survey paper of Blass in Handbook of Set Theory), "small uncountable cardinals" also is a (part of the) title of an influential paper by Vaughan (pdfs.semanticscholar.org/9065/…) in "Open problems in Topology". Maybe the problem is that Vaughan is mainly a set-theoretic topologists and set theorists percept this topic a bit differently.
@YCore Anyway "small" relates to cardinalities of objects appearing in Classical Matehmatics: not very far from the continuum. If you try to label each natural mathematical object with its complexity in the Comulative Hierarchy of the Universum, you will discover that the most difficult objects of Functional Analysis and Differential Equations (like functions spaces of spaces of operators) have complexity not more $\omega+13$ (maybe $\omega+15$). Which means that the cardinality of such natural objects is at most $\beth_{15}$. This can be informally considered as the realm of small cardinals.
"infinite-combinatorics" descibes rather methods and results involving combiknatorial properties of infinite sets. It is situated somewhere in the union of Set Theory and Combinatorics. Eventually this tag also can be created to point at the essense of a problem (which concerns combinatorial properties of infinite sets).
So, up to my understanding of the realm of uncountable cardinals it can be divided into three categories: "small" (those in the interval $[\omega_1,\mathfrak c]$, "intermediate" (larger than $\mathfrak c$ but existing in ZFC, and finally "large" -- which do not necessarily exist in ZFC.
Thank you for supporting the idea of introducing the tag "small-uncountable cardinals". Concerning the boundary between small and large cardinals is very precise: the existence of small cardinals follows from ZFC whereas the existence of large cardinals (starting with strong inaccessible) is a stronger assumption than the consistency of ZFC. More precisely this is a boundary between large cardinals and non-large. The boundary between small and non-small is $\mathfrak c$, the cardinality of continuum.
