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incorporated Paul Fabel's comment into the question in an attempt to correct the definition of length space
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Ricardo Andrade
Ricardo Andrade
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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?

(RecallEdit 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 $f:[0,d(x,y)] \to X$ such that$x$ to $t-e< d(x,f(t))+d(f(t),y)< t+ e$, for all$y$ whose length is less then $t \in [0,d(x,y)]$$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?

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

(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 path $f:[0,d(x,y)] \to X$ such that $t-e< d(x,f(t))+d(f(t),y)< t+ e$, for all $t \in [0,d(x,y)]$.)

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?

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?

fixed and added tex/mathjax displays
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Ricardo Andrade
Ricardo Andrade
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Does every connected metrizable locally path connected topological space X$X$ admit a compatible metric d$d$ so that (X,d)$(X,d)$ is a length space?

(Recall thethat a metric space (X,d)$(X,d)$ is a length space if for every x$x$ and y$y$ in X$X$ and every $e$>0$e>0$, there exists a path $f:[0,d(x,y)] --> X$$f:[0,d(x,y)] \to X$ such that t-e< d(x,f(t))+d(f(t),y)< t+ e$t-e< d(x,f(t))+d(f(t),y)< t+ e$, for all t in [0,d(x,y)]$t \in [0,d(x,y)]$.)

The answer is certainly `yes'yes for Peano continua, (butbut this is not a trivial fact).

More generally the answer is apparently `yes'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?

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

(Recall the metric space (X,d) is a length space if for every x and y in X and every $e$>0, there exists a path $f:[0,d(x,y)] --> X$ such that t-e< d(x,f(t))+d(f(t),y)< t+ e, for all t in [0,d(x,y)].)

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?

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

(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 path $f:[0,d(x,y)] \to X$ such that $t-e< d(x,f(t))+d(f(t),y)< t+ e$, for all $t \in [0,d(x,y)]$.)

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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Paul Fabel
Paul Fabel
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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?

(Recall the metric space (X,d) is a length space if for every x and y in X and every $e$>0, there exists a path $f:[0,d(x,y)] --> X$ such that t-e< d(x,f(t))+d(f(t),y)< t+ e, for all t in [0,d(x,y)].)

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?