TITLE: Idea behind pattern seen in topology
QUESTION [7 upvotes]: In topology, I have come across a "pattern" that involves a subsequence or subset of a sequence or set. For example, this definition of a second countable space uses:

A topological space $T$ is second countable if there exists some countable collection $\mathcal{U} = \{U_i\}_{i=1}^{\infty}$ of open subsets of $T$ such that any open subset of $T$ can be written as a union of elements of some subfamily of $\mathcal{U}$.

Another variant of that "pattern" can be found in:

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Then $\lim_{n\to\infty}x_n = L.$

What is the idea behind that pattern of a subsequence/subset of sequence/set? I realize that the two examples are different, I am not saying that they are the same, but it feels to me that they share an important idea of "a sub-thing of a thing does this or that" in their construction, and it is that idea/thought process that I'd like to get at. 
As a bonus question: in which fields/sub-fields of maths is that idea more prevalent/used? Does it show up only in e.g (some branch of) topology, or is it also used in e.g. (some branch of) algebra?

REPLY [2 votes]: A topological space $T$ is second countable if there exists some countable collection $\mathcal{U} = \{U_i\}_{i=1}^{\infty}$ of open subsets of $T$ such that any open subset of $T$ can be written as a union of elements of some subfamily of $\mathcal{U}$.

The property second countability of a topological space $T$ is characterized by having a countable base.
The observed pattern here is a realisation of the fundamental principle of searching for basic building blocks of mathematical structures. We can also see this pattern e.g. when looking at vector spaces $V$ and bases of $V$. 

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Then $\lim_{n\to\infty}x_n = L.$

Although the pattern here looks similar as above, I think the principle behind it is somewhat different. Bounded sequences are nice, but usually convergent sequences are much nicer to work with and it is natural to ask: What additional properties are necessary for a bounded sequence to be convergent?
In this case we are not looking for basic building blocks as above, but instead for a property shared by all specific sub-structures (i.e. all convergent subsequences converge to $L$). 

Note: I think identifying such patterns is great and essential for a fruitful mathematical development. But we also should  carefully check  interpretations which come into mind before accepting them.
Compactness and second countability are quite different concepts in general topological spaces. There are topological spaces for each of the four combinations: compactness (yes/no) and second countability (yes/no). Another aspect indicating a fundamental difference is that the uncountable product of a compact topological space is compact, whereas the uncountable product of a second countable space need not be second countable.
Hint: A great resource to look for topological spaces having specific properties and to look for topological properties and their relationship is Counterexamples in Topology by L.A. Steen and L.A. Seebach.