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Let $A$ be a set. Then it seems common to name cover of $A$ any set $\mathcal C$ of sets with $\bigcup{\mathcal C}\supseteq A$.

  1. Is there then a good name for the particular covers $\mathcal C$ with $\bigcup{\mathcal C}=A$ ?

  2. Are there standard names for those covers $\mathcal C$ with $\emptyset\notin\mathcal C$ ?

Question 2 also arises for partitions. Well-regarded books allowing the empty set to be a member of a partition (which I then would only call a quasipartition) but using the term differently later on include Kelley's and Dugundji's topology texts.

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The construction "covering of $A$ by [something]" seems handy. For (1), one can use "covering of $A$ by subsets of $A$", and for (2) one can use "covering of $A$ by nonempty sets".

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I was hoping for something shorter. – Lutz Mattner Mar 7 '13 at 20:29

An answer to the first question is: An exhaustion of $A$.

Before I asked, I really thought for a while, browsed a few books, and did some googling, but somehow this standard term just did not occur to me then. Im wondering how many of the 120 or so viewers of the question just tried to be polite in not answering.

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Perhaps those 120 people also had not heard of this "standard" term. I would suggest that if you use it, you include the definition, otherwise your reader might grow tired of looking for the term, just like you did. Searching for exhaustion sets brings up as many references to weightlifting as it does to set theory. – Zack Wolske Mar 7 '13 at 21:57

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