TITLE: are "all nets in $X$" well defined?
QUESTION [5 upvotes]: Denote $f:X\to Y$ as a function between topological spaces $X$ and $Y$. One good way for determining whether $f$ is continuous is to check the following statement.

$f$ is continuous iff for every converging net $(x_\alpha)_{\alpha \in \Lambda}$ in $X$, one has $\lim_\alpha f(x_\alpha) = f(\lim_\alpha x_\alpha)$.

However, "every converging net" seems not to be a well defined set, since the index set $\Lambda$ could in principle be any directed set with any cardinality. I wonder what one needs to give this a well defined meaning. From what I read, Wikipedia says nothing about it.

REPLY [3 votes]: Additional remark ... The first textbook to embrace the use of nets for general topology was General Topology by J. L. Kelley.  It has an appendix "Elementary Set Theory" ... if examined carefullly we see its use of classes is more extensive than the NBG set theory.  It has subsequently been studied on its own as Morse-Kelley Set Theory:

Morse–Kelley set theory is a proper extension of ZFC. Unlike von Neumann–Bernays–Gödel set theory, where the axiom schema of Class Comprehension can be replaced with finitely many of its instances, Morse–Kelley set theory cannot be finitely axiomatized.  

So, for example, in Kelley we can construct the completion of a uniform space $X$ by starting: take the Class of all Cauchy nets in $X$, then form a Quotient by a certain equivalence Relation... [capitalized a la Conway]
In any given instance we can re-do the argument in ZF-style.  But Kelley (I guess) is saying, "Why bother with that?"