@AsafKaragila thanks!

@AsafKaragila hi, may I ask which paper of yours implies this result? I would like to read it

@HanulJeon $F(\xi)$ is a fixed value, but the closure of $F[\kappa\setminus\lambda]$ under $F$ will be $\bigcup\{F(\xi),F(F(\xi)),...\}$, which will be unbounded(where $\xi$ is some arbitrary value $>\lambda$)

"fix some cofinal $(λ_i\mid i\in\omega)$, and for $k$ be the minimal $k$ such that $x∈λ_k$, map $x$ to some element of $\lambda_{k+1}\setminusλ_k$" If $\kappa=ω_{ω+1}$ and $λ=ω_ω$, send all elements in $\kappa\setminusλ$ to some $x$. If $x\inω$, then $F(x)∈ω_1$, if $x∈ω_1$ then $F(x)∈ω_2$ etc. Then the closure of every subset of $λ$ under $F$ is cofinal

@HanulJeon but we are looking at the closure of the image, not just the image itself

@JohannesSchürz doesn't similar thing works for $\kappa$ weakly compact? Regardless on $λ$

@AsafKaragila I see, anyhow thanks for pointing it out, next time I will use $\sf ZF$-regularity from the start

@AsafKaragila New edition