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Apr 28 at 1:48 comment added bof @DominicvanderZypen It is not exactly the existence of a "weakly minimal cover" but a slightly stronger proposition, namely the Lemma in my partial answer to your follow-up question, which can be used to show that every linear cover of a countable set has a minimal subcover.
Apr 28 at 1:34 history edited bof CC BY-SA 4.0
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Apr 26 at 13:55 history edited bof CC BY-SA 4.0
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Apr 26 at 13:34 comment added Iosif Pinelis @bof : In the OP, $\operatorname{cov}_{\mathcal C}(x)$ was defined as a number. In your answer, $\operatorname{cov}(x)$ seems to act a set.
Apr 26 at 13:20 history edited bof CC BY-SA 4.0
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Apr 26 at 12:46 comment added Iosif Pinelis What is $\operatorname{cov}(x)$?
Apr 26 at 8:54 comment added Dominic van der Zypen Brilliant, thanks @bof. I suppose this can also be used to show that every linear cover contains a minimal cover
Apr 26 at 8:54 vote accept Dominic van der Zypen
Apr 26 at 8:47 history answered bof CC BY-SA 4.0