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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?

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I'm not sure exactly what you're looking for, but both of these concepts seem to me to indicate some idea of compactness. Compactness becomes important in analysis because it is preserved through continuous images, i.e. the continuous image of a compact set is compact. This is important because it allows us to show for example if $f$ is continuous and $x_n \in A$ where $A$ is compact, that while showing $f(x_n)$ converges does not imply that $x_n$ converges, since $A$ is compact, $x_n$ must have a convergent subsequence $x_{n_k} \to z$, and as a consequence of continuity, $f(x_n)$ will converge to $f(z)$. In general compact sets are often a lot nicer to work with than other sets because of some of these subsequential and subset properties.

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    $\begingroup$ I think that is the kind of connection I am looking for. In the mind of the topologist, seeing such a pattern of a sub-thing of thing doing this or that probably triggers ideas of compactness - this probably makes discussions between topologists simpler because this pattern is well-known as related to compactness. $\endgroup$
    – Frank
    Apr 18, 2017 at 19:38
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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.

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Here's one thing I've noticed. In topology, questions about cardinality become very important. Here's an extremely simple idea to demonstrate this: a union of finitely many closed sets is closed, but a union of infinitely many closed sets may not be closed. Already, we have so many problems.

This is why compactness is such an important property. How do you connect the infinite to the finite? If you have infinitely many opens sets doing something, you want ONLY finitely many of them to do the job.

A nontrivial example of this would be to consider a continuous function $f:X\rightarrow Y$ of metric spaces. If $X$ is compact, then $f(X)$ is compact, so for any infinite open cover there is a finite open subcover, and if we pretend that we cover $f(X)$ by infinitely many bounded open sets, we only need finitely many of them to do the job, so $f(X)$ itself is bounded. This is the basis for the extreme value theorem to show $f$ has a maximum and minimum.

I can't think of anything else off the top of my head, but I can also say that the question of countable vs uncountable is probably important too.

In abstract algebra (at least at the elementary level), the question of cardinality isn't so much important, so it's no surprise that I don't see this theme showing up there.

Actually, now that I think about it, I think category theory is a good example where you relate things to their sub-things.

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  • $\begingroup$ Thanks! See my comment to @awrigh96. $\endgroup$
    – Frank
    Apr 18, 2017 at 19:38
  • $\begingroup$ can you elaborate on the connection between finite and infinite? Also, in my second example, the sub-thing is not necessarily finite, or countable, but rather convergent. Yet we still have some property on a sub-thing of the thing, still the "same pattern"? $\endgroup$
    – Frank
    Apr 18, 2017 at 19:42
  • $\begingroup$ I'm not sure if I can elaborate. I was just saying that the property of compactness itself is what connects the finite and infinite, which is why we care about it. I guess what I was trying to get at was that different things occur at different carindalities and that's useful to exploit. In general, however, a property may be more obvious if you look at the subcomponents of a thing and that's probably the main point. Not sure if that's obvious or what, but yeah. $\endgroup$ Apr 18, 2017 at 20:08
  • $\begingroup$ Ahh, nvm. I deleted another comment of mine because I misread a part of the post. $\endgroup$ Apr 18, 2017 at 20:12
  • $\begingroup$ I read somewhere that rather than cardinalities, the niceness of compact spaces was that things that were true locally could be true globally on the whole space. Is that what you had in mind? $\endgroup$
    – Frank
    Apr 18, 2017 at 21:10
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There are many cardinals associated with a given topology $T$ on a set $X$. These are called topological cardinal functions.

Examples: The weight $w(T)$ is the least infinite cardinal $k$ such that $T$ has a base $B$ with $|B|\leq k$. Second-countability of $T$ means $w(T)=\aleph_0.$ The density $d(T)$ is the least infinite cardinal $k$ such that $X$ has a dense subset $Y$ with $|Y|\leq k$. Separability means $d(T)=\aleph_0$. The cellularity $c(T)$ is the least infinite cardinal $k$ such that if $F$ is a family of pair-wise disjoint open subsets of $X$ then $|F|\leq k.$

There are many others. It is common to write $f(X)$ instead of $f(T$) when $f$ is a top'l cardinal function of $T.$ And we define $hf(X)=\sup\{ f(Y):Y\subset X\}.$ (The $h$ is for "hereditary").

We always have $hw(X)=w(X)\geq d(X)\geq c(X)$ but there are other top'l cardinal functions that do not fall into this pattern.

For a metric space $X$ we have $w(X)=d(X)=hd(X)=c(X)=hc(X).$ This is useful for proving many things about open or closed subsets of $\mathbb R^n .$

Countability is important in Measure Theory, e.g. in the development of Lebesgue measure, and much of its use is from the fact that so many top'l cardinal functions on $\mathbb R^n$ have the value $\aleph_0$.

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