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Offered a range like we’d like to find out if there’s a pure way to decompose it into its “components” . These aren’t its connected elements while in the topological feeling, but in almost any acceptable perception The 2 pieces are unrelated except possibly exactly where they intersect.
It follows that there's a sequence of pairwise disjoint arcs in $Y$ which converges to $Y$ during the Hausdorff length. By Theorem 30 in Paper B, $Y$ is non-Suslinean.
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The one thing still left to confirm is always that $Y=X/sim$ (from Instance 3) is Hausdorff, to ensure that $Y$ is actually a metrizable
The “obvious” way to determine homomorphisms for a significant course of objects includes thinking about them as “sets with additional structure,” after which you can homomorphisms are capabilities that protect that additional composition.