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Does every connected metrizable locally path connected topological space $X$ admit a compatible metric $d$ so that $(X,d)$ is a length space?

(Edit to correct definition: Recall that a metric space $(X,d)$ is a length space if for every $x$ and $y$ in $X$ and every $e>0$, there exists a rectifiable path from $x$ to $y$ whose length is less then $d(x,y)+e$.)

The answer is certainly yes for Peano continua, but this is not a trivial fact.

More generally the answer is apparently yes for such locally compact spaces, but local compactness is certainly not necessary: for example, familiar Hilbert space is a length space.

Do the above claims survive without local compactness?

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There is something wrong with the inequality on $f$. Take $t=0$ for example. –  Sergei Ivanov May 14 '13 at 9:54
Thanks Sergei! My obtuse (and likely incorrect) definition should be ignored. The reader should use any standard definition of length space. For example, for all x and y in X and all e>0, there exists a rectifiable path from x to y whose length is less then d(x,y)+e. –  Paul Fabel May 14 '13 at 21:28
@Paul: I have incorporated your comment into the statement of the question, so as to hopefully correct the definition. –  Ricardo Andrade Jul 8 '13 at 21:54
Thanks Ricardo! –  Paul Fabel Aug 27 '13 at 23:12
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1 Answer

This is not an answer but a long comment with references for the locally compact case. It is proved in [Tominaga-Tanaka, Convexification of locally connected generalized continua. J. Sci. Hiroshima Univ. Ser. A. 19 (1955), 301–306] that

Theorem. Every locally compact, connected, locally connected, separable metrizable space admits a complete length metric.

The proof is similar to the one in the compact case which is due to Bing and Moise (independently). A relative (stronger) version is due to Doodley [Further extending a complete convex metric. Proc. Amer. Math. Soc. 40 (1973), 590–596] and [Extending a complete convex metric. Proc. Amer. Math. Soc. 34 (1972), 553–559] whose electronic versions can be easily found on the web.

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