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On a topological space $X$, a net is defined as a map $A \ni \alpha \longmapsto x_{\alpha} \in X$ from a directed set $A$.

With this, compactness of $X$ (for instance) is equivalent to "every net $(x_\alpha)$ in $X$ has a subnet with a limit in $X$".

Does is suffice (not only for compactness) to consider equivalent classes of isomorphic directed sets up to the cardinality of the set $X$ -- which might be a set?

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    $\begingroup$ We quantify over things which are not sets all the time in mathematics - why do you think there's something problematic about it? Almost always when you write a universal property you do this kind of quantification, e.g. "for every set $Z$ with maps $Z \to Y$ and $Z \to X$ there is a unique map $Z \to X \times Y$...", here we say "for all $Z$" where $Z$ runs over the proper class of all sets. $\endgroup$
    – Dan Petersen
    Commented Apr 26, 2019 at 8:28
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    $\begingroup$ @DanPetersen Quantifying over the whole universe is fine, but it may still be interesting to know of a small set of nets that suffice in testing a space $X$ for compactness. $\endgroup$
    – bof
    Commented Apr 26, 2019 at 10:19
  • $\begingroup$ Yeah sure. Giving it a second thought, there should be no problem. Usually, quantfiers are unbounded anyway. But the second question keeps still there and is worth thinking about, I guess. $\endgroup$
    – mjungmath
    Commented Apr 26, 2019 at 10:35

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