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  1. $X$ is not a separable submetrizable, i.e.($iw(X)>\omega$) $X$ has not a countable injective weight.
  2. There is a Baire isomorphism 1-class between $X$ and a separable metrizable space $Y$.
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Now take any uncountable $Q$-set $Y\subset R$ and let $X$ be the one-point compactification of a discrete space of cardinality $|Y|>\omega$.

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  • $\begingroup$ Then $X$ and $Y$ is Borel isomorphic (1 - class), but $X$ and $Y$ is not Baire isomorphic ! So the question remains open ! $\endgroup$
    – Alexander Osipov
    Commented Nov 8, 2017 at 6:33

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