are "all nets in $X$" well defined? 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.
 A: 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?"
A: (This answer echoes Michael Greinecker's comment above)
There is not set of all convergent nets in a space, but this does not mean that this characterisation of continuity is flawed in any real way.  Certainly we can recognise whether or not some object is a convergent net in a the topological space $X$, and then verify that the required condition holds.  This would be done in the usual mathematical way: start with an arbitrary (and unspecified) convergent net $( x_\alpha )_\alpha$ in $X$, and by the properties of the spaces $X , Y$ and the function $f$ show that if $x$ is any limit point of the net then $f(x)$ is a limit point of the net $( f ( x_\alpha ) )_\alpha$.
This is not unusual in mathematics.  Consider the universal property of free groups: 

Given any set $S$, the free group generated by $S$ is the unique (up to isomorphism) group $F_S$ having $S$ as a subset such that for any group $G$ and mapping $f : S \to G$ there is a unique homomorphism $\varphi : F_S \to G$ such that $f ( s ) = \varphi ( s )$ for all $s \in S$.

Although there is certainly no set of all groups, we can still verify that a group satisfies the universality property by takings an arbitrary (and unspecified) group $G$ and an arbitrary (and unspecified) function $F : S \to G$ and show that the required group homomorphism $\varphi : F_S \to G$ exists and is unique.
