Let $X$ be a metric continuum (compact + connected) which is the one-to-one continuous image of the interval $[0,\infty)$. Such an $X$ is called a linear continuum.


It seems like $X$ should be chainable (as defined in the 2nd paragraph [here](http://www.auburn.edu/~mincpio/pm/newextension.pdf)). Equivalently, for every $\epsilon>0$ there should be a mapping $f:X\to [0,1]$ such that the pre-image of each point in $[0,1]$ has diameter of $<\epsilon$. This property is also discussed here: [Beautiful examples of arc-like continua](https://mathoverflow.net/questions/112317/beautiful-examples-of-arc-like-continua).


Is this true, and it so why?



Continuum image of line is chainable?
The circle is a linear continuum according to the definition provided in the question. But the circle is not chainable, by the Borsuk-Ulam Theorem.


