Mapping one tree onto another

Question

Let $T$ and $T'$ be trees (finite acyclic graphs in the obvious compact connected topology).

Let $f:T\to T'$ be a continuous surjection.

Let $A'$ be an arc in $T'$ (by arc I mean $A'\simeq [0,1]$). Is there necessarily a tree or an arc in $T$ which $f$ maps onto $A'$?

Answer 1

no. take these two tress: a line segment, and a tree with 3 leafs, like this:

        a -----o ----- c
               -
               -
               -
               b

define f as such: start from a, go to o, go to b, return to o, go to c.

the line segment a-c dosn't satisfy your condition.