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Post Closed as "Not suitable for this site" by Benoît Kloeckner, Tom LaGatta, Michael Greinecker, Ramiro de la Vega, Daniel Moskovich
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Tom LaGatta
Tom LaGatta
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Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?

If not, is $f$ a measurable isomorphism from $X$ to $Y$?

Let $X$ and $Y$ be topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?

If not, is $f$ a measurable isomorphism from $X$ to $Y$?

Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?

If not, is $f$ a measurable isomorphism from $X$ to $Y$?

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Tom LaGatta
Tom LaGatta
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Let $X$ and $Y$ be topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?

If not, is $f$ a measurable isomorphism from $X$ to $Y$?

Let $X$ and $Y$ be topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?

Let $X$ and $Y$ be topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?

If not, is $f$ a measurable isomorphism from $X$ to $Y$?

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Tom LaGatta
Tom LaGatta
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Extending a homeomorphism from a dense set

Let $X$ and $Y$ be topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ is a homeomorphism from $D$ to $f(D)$.

Does it follow that $f$ is a homeomorphism from $X$ to $Y$?