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clique concept – Is the Rudin-Keisler ordering a steady relation? Answer

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clique concept – Is the Rudin-Keisler ordering a steady relation?

If $X, Y$ are topological, and $Rsubseteq Xtimes Y$ we are saying that $R$ is steady (from $X$ to $Y$) if $$R^{-1}(V) = {uin U: exists vin V:(u,v)in R}$$ is launch in $X$.

Let $textual content{NPU}(omega)$ breathe the clique of non-principal ultafilters on $omega$. The Rudin-Keisler preorder on $textual content{NPU}(omega)$ is outlined by
$${cal U} leq_{RK} {cal V} :Leftrightarrow (exists f:omegatoomega)(forall Uin{cal U}) f^{-1}(U)in {cal V} $$ for ${cal U}, {cal V}in textual content{NPU}(omega)$. It is simple to behold that $leq_{RK}$ is reflexive and transitive, however not anti-symmetric.

Let $[omega]^omega$ denote the gathering of innumerable subsets of $omega$. For $Ain[omega]^omega$, let $$u(A) = {{cal U}in textual content{NPU}(omega): Ain {cal U}}.$$ On $textual content{NPU}(omega)$ we deem the topology generated by the the gathering ${u(A): Ain[omega]^omega}$.

Question. Is $leq_{textual content{RK}}$ a steady relation on $textual content{NPU}(omega)$ with this topology?

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