Understanding the definition of monotonically monolithic A collection $\mathcal{N}$ of subsets of $X$ is called an external network of $A \subset X$, when for every $x \in A$ and every neighbourhood $U$ of $x$, there exists some $N \in \mathcal{N}$ such that $x \in N \subset U$. Note that in that case $\left\{ N \cap A: N \in \mathcal{N} \right\}$ is a network for the subspace $A$ (where a network is like a base, except that its members need not be open sets), so it's like a network for $A$ except that they are allowed to "stick out of" $A$. Also note that the monotonic condition wouldn't make sense if we wanted networks of the sets, instead of external networks.
Now, a space $X$ is called monotonically monolithic when for every subset $A$ of $X$ we assign an external network $\mathcal{O}(A)$ of $\overline{A}$, such that the following conditions hold:
1) $\left| \mathcal{O}(A) \right| \le \max\{|A|,\omega\}$ for all $A \subset X$.
2) For all $A \subset B \subset X$, $\mathcal{O}(A) \subset \mathcal{O}(B)$.
3) For all ordinals $\alpha$ and families (of subsets of $X$) $(A_\beta)_{\beta < \alpha}$ such that $\beta < \beta' < \alpha$ implies $A_{\beta'} \subset A_\beta$, then $\mathcal{O}(\cup_{\beta < \alpha} A_\beta) = \cup_{\beta < \alpha} \mathcal{O}(A_\beta)$.
Here we can see condition (1) witnesses that "monolithic", and condition (2) witnesses "monotonically". 

My quesyion is that what's the intention of the author for adding the condition (3)?

 A: Look at proofs about this property and where they use it in their proofs. Is it essential, or just convenient? Also, suppose we have operator $\mathcal{O}$ that satisfies (1) and (2). If we have a family $A_\beta, \beta < \alpha$ that is monotonic as in (3) [corrected] then (2) already implies $\mathcal{O}(A_\beta) \subset \mathcal{O}(\cup_{\beta < \alpha} A_\beta)$ and so $\cup_{\beta < \alpha} \mathcal{O}(A_\beta) \subset \mathcal{O}(\cup_{\beta < \alpha} A_\beta)$, already. So the equality in (3) shows that the external network for the union of the $A_\beta$ is sort of minimally economical: it's as large as it needs to be and no larger.
But again, the proof of the pudding is in the eating. Probably, in the "natural" examples the third property sort of follows naturally. Maybe it simplifies proofs, or maybe is even essential in those proofs. Like Brian, I haven't studied it at all. (My thesis was called "monolithic hyperspaces", so I know about monolithic spaces, which are quite interesting...)   
A: You have the inclusion backward in (3): you want $A_\beta\subseteq A_{\beta\,'}$ when $\beta<\beta\,'<\alpha$. 
I suspect that one would have to work with the property for a while to decide how useful or important condition (3) is, and I’ve not done so. It’s a very natural condition, however: it says that the operator $\mathcal{O}$ is not just monotone, but is also continuous in a common set-theoretic sense of the word. You can also look upon it as saying that the external network assigned to an infinite set $A$ is completely determined by the external networks assigned to proper subsets of $A$, since $A$ can always be written as an increasing union of proper subsets indexed by a limit ordinal. At the very least this makes $\mathcal{O}$ much easier to work with.
