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Martin Sleziak
Martin Sleziak
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A famous theorem on space-filling curves is the Hahn-Mazurkiewicz theoremHahn-Mazurkiewicz theorem: Let X$X$ be a Hausdorff space, then there exists a surjective continuous map [0,1] \to X$[0,1] \to X$ if and only if X$X$ is compact, connected, locally connected and metrizable.

Is there a similar characterisation for all (Hausdorff) spaces having a surjective continuous map into the unit interval (which I decided to call line-filling spaces)?

A famous theorem on space-filling curves is the Hahn-Mazurkiewicz theorem: Let X be a Hausdorff space, then there exists a surjective continuous map [0,1] \to X if and only if X is compact, connected, locally connected and metrizable.

Is there a similar characterisation for all (Hausdorff) spaces having a surjective continuous map into the unit interval (which I decided to call line-filling spaces)?

A famous theorem on space-filling curves is the Hahn-Mazurkiewicz theorem: Let $X$ be a Hausdorff space, then there exists a surjective continuous map $[0,1] \to X$ if and only if $X$ is compact, connected, locally connected and metrizable.

Is there a similar characterisation for all (Hausdorff) spaces having a surjective continuous map into the unit interval (which I decided to call line-filling spaces)?

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skupers
skupers
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Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces?

A famous theorem on space-filling curves is the Hahn-Mazurkiewicz theorem: Let X be a Hausdorff space, then there exists a surjective continuous map [0,1] \to X if and only if X is compact, connected, locally connected and metrizable.

Is there a similar characterisation for all (Hausdorff) spaces having a surjective continuous map into the unit interval (which I decided to call line-filling spaces)?