The frechet filter on an uncountable cardinal $\kappa$

Question

How can I see that any unbounded (not necessarily closed) set of a cardinal $\kappa$ meets (has a nonempty intersection) with a set $A$ in the Frechet filter on $\kappa$ i.e. $A$ has the property $\kappa\setminus A<\kappa$ ?

Answer 1

First, a comment on terminology: Your definition of the Fréchet filter is non-standard. Usually, the Fréchet filter on a set $X$ is the set of all cofinite subsets of $X$: $$\{A\subseteq X\mid X\setminus A\text{ is finite}\}.$$ I don't know if there's a standard name for your filter $$\mathcal{F} = \{A\subseteq \kappa\mid |\kappa\setminus A|<\kappa\},$$ so I'll just call it $\mathcal{F}$.

Second, the way the question is worded is a bit ambiguous. Do you want every unbounded set to meet every set in $\mathcal{F}$ or just some set in $\mathcal{F}$? Of course, the latter interpretation is trivial, since $\kappa\in \mathcal{F}$ and every unbounded set meets $\kappa$. So I'll assume that you want to show that every unbounded set in $\kappa$ meets every set in $\mathcal{F}$.

Ok, but note that a set $U\subseteq \kappa$ meets every set in $\mathcal{F}$ if and only if $|U| = \kappa$. Indeed, if $|U| = \kappa$, then for any $A\in \mathcal{F}$, $U\not\subseteq (\kappa\setminus A)$, so $U\cap A \neq \emptyset$. Conversely, if $|U|<\kappa$, then $U$ fails to meet $\kappa\setminus U\in \mathcal{F}$.

So every unbounded set in $\kappa$ meets every set in $\mathcal{F}$ if and only if every unbounded set in $\kappa$ has cardinality $\kappa$. And this is just the definition of regularity of $\kappa$.