Testing points on the closure of a union of increasing subsets

Question

Let $X$ be a topological space. Let $(S_\lambda)_{\lambda\in\Lambda}$ be a net of closed subspaces of $X$, and suppose that for all $\lambda\le\mu$, $S_\lambda\subseteq S_\mu$. Let now $$ x\in \mathrm{closure} \left( \bigcup_{\lambda\in\Lambda} S_\lambda \right) . $$

Can we express $x$ as a limit of a net $(x_\lambda)_{\lambda\in\Lambda}$ where for each $\lambda$, $x_\lambda\in S_\lambda$?

Answer 1

No: Consider $\beta \mathbb N$, the Stone-Cech compactification of $\mathbb N$.
Pick $ x \in \beta \mathbb N \setminus \mathbb N$. For $n \in \mathbb N$ define $S_n = \{1, ... n\}$, which is a closed subspace of $\beta \mathbb N$.
$x \in \text{cl}(\mathbb N) = \text{cl}(\cup_{n \in \mathbb N} S_n)$.
However, there is no sequence in $\mathbb N$, which converges to $x$.