Definition of a closed set

Question

In my set theory notes, when defining club sets, of course we first defined what does it mean for a set to be closed.

Namely, the definition is following,

a subset $C \subset \kappa$ is said to be closed in $\kappa$ if $\forall \lambda < \kappa($ $\lambda$ is a limit ordinal and $C\cap\lambda$ is unbounded in $\lambda \to \lambda \in C)$, but I find this definition very hard for intuitive understanding.

Can someone give me some further explanation?

Answer 1

Basically, the idea is that if $C\cap \lambda$ is unbounded in $\lambda$, then $C$ has a "sequence" of elements approaching $\lambda$; so if $C$ is closed, we expect every limit of such a "sequence" to be in $C$.

I put the word "sequence" in quotes above since it's not exactly correct - "sequence" usually means "sequence of ordertype $\omega$, like $a_1, a_2, . . .$. Here, our "sequence" could have length up to $\lambda$! The better term to use here would be net, but for intuitive purposes I wanted to use "sequence", even though it's technically incorrect.

This in fact agrees with the topological notion of closedness: there's a natural topology on a linear order, and under this topology the closed sets are exactly the closed sets (you know what I mean :P).

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EDIT: There's a second piece to this: why do we only consider "sequences" approaching $\lambda$ from below?

Well, this is because there aren't any approaching $\lambda$ from above! There's a limit to how close to $\lambda$ you can get, from above: no closer than $\lambda+1$. There's no ordinal $>\lambda$ which is $<\lambda+1$. So the only way to "approach" $\lambda$, is from below - which is why it's enough to ask whether $C\cap \lambda$ (the part of $C$ below $\lambda$) is unbounded in $\lambda$ ("approaches" $\lambda$).

Answer 2

An unbounded set $C \subset κ$ is closed if and only if for every sequence $α_0 < α_1 < \cdots < α_ξ < \cdots $ $(ξ < γ)$ of elements of $C$, of length $γ < κ$, we have $lim_{ξ \rightarrow γ} α_ξ \in C$. This is equivalent to $C$ es closed in the order topology of $\kappa$.