The main Idea here is that sequence convergence in not first countable spaces is not enough to explain closure and continuity. I want to discuss this with two examples:
(1) Lets have $\mathbb{R}^{\mathbb{R}}$ with the Topology of pintwise convergence and lets have the subset
$ E:=\{g\in \mathbb{R}^{\mathbb{R}}|g(x)\neq 0 \textit{ only for finite many } x\}$
Lets have than the constant function $f(x)=1$ with $x\in \mathbb{R}$. Now it is $f\in cl(E)$ but there is no sequence that has $f$ as limit because of the definition of $g$.
(2) Lets $\Omega=[0,\omega_1]$ and $\Omega_0=[0,\omega_1)$. Here we have $\omega_1\in cl(\Omega_0)$ but aswell no sequence in $\Omega_0$ convergent to $\omega_1$.
Based on this there should be
(1) A net in $E$ convergent to $1$.
(2) A net in $\Omega_0$ convergent to $\omega_1$
How can I approach such problems? Is there a way to find such a net for each of them?
What works for any space, including these: Let $x \in \overline{E}$ and let the directe set for the net be $\mathcal{O}_x$, the set of open neighbourhoods of $x$, directed by $O \le O' \iff O' \subseteq O$ (smaller neighbourhoods as sets are bigger, the intersection of two neighbourhoods is a common upper bound).
For each $O \in \mathcal{O}_x$ pick $f(O) \in O \cap E$. This $f: \mathcal{O}_x \to X$ is is a net in $E$ that converges to $x$. This is clear from the definitions.
For $(2)$ we can also use $\Omega_0$ as the directed set and the identity as the net. Nice and efficient, and no choice required.
For $(1)$ and alternative net is : the finite subsets of $\Bbb R$ as directed set $F$ with inclusion as direction, and $x_F$ the point that is $1$ on $F$ and $0$ outside of $F$. Some thought will reveal that this converges to the constant $1$ function.
Both are really the standard (first one) in disguise, but made more explicit.