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2
votes
Proper coverings as a quotient of the coverings
I don't think they are isomorphic. Let $X=\mathbb N$ and let $A$ be a covering consisting of the sets of the form
$$ a_n = \{ 0, \dots, n \} $$
for every $n\in \mathbb N$.
Then, $A$ is not $\simeq$-e …