# Is every connected metrizable locally path connected space a length space?

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