Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map $i$?
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asked Apr 21, 2020 at 20:36
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1$\begingroup$ Restatement: let $Z$ be a topological space homeomorphic to a dense subset of a locally compact space. Let $X$ be a locally compact space. Let $f$ be a continuous bijection $X\to Z$. When is $f^{-1}:\mathrm{Borel}(Z)\to\mathrm{Borel}(X)$ bijective? (it is clearly injective) $\endgroup$– YCorCommented Apr 21, 2020 at 20:49
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2$\begingroup$ @YCor "homeomorphic to a dense subset of a locally compact space" - that's just a Tychonoff space $\endgroup$– erzCommented Apr 21, 2020 at 22:49
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$\begingroup$ If $X$ and $Y$ are second countable, then they are Polish spaces, so $i$ is a Borel isomorphism onto its image by a well-known corollary of a theorem due to Lusin and Souslin (see Corollary 15.2 of Kechris's Classical Descriptive Set Theory). In fact, I am only using the local compactness to prove that $X$ and $Y$ are regular so I can apply Urysohn's metrization theorem. Outside this case, I don't think there are any useful criteria. First countability is not helpful, by the example of $\mathbb{R}$ mapping onto itself where as $X$ it has the discrete topology and as $Y$ it has its usual one. $\endgroup$– Robert FurberCommented Apr 22, 2020 at 21:12
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