In the paper of Norman Howes "[A note on transfinite sequences](https://eudml.org/doc/211097)" is mentioned that Miscenko space 

$M = \{f \in \prod_{n \in \omega \smallsetminus \{0\}} \aleph_n+1 | \space \exists k \space  \space \forall m \geq k \space f(m) < \aleph_m\}$

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is not normal. The closed sets that support this are:

$H = \{f \in M | \space \forall k \space f(k) \neq 0 \} $.

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$K = \{f \in M | \space \exists k \space f(k+1)=0 \space \space and \space \forall m \leq k \space f(m) = \aleph_m\} $

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How to prove that for any open $U$, $V$ such that $H \subset U$ and $K \subset V$,   $U \cap  V \neq \emptyset$ ?