# Two arcs equal three arcs

(Gonna answer my own question, as is encouraged.)

To set the stage: an arc (or a Jordan arc) is a non-self-intersecting curve with two distinct endpoints. (For those who are familiar with topology, it's a subset of the plane homeomorphic to $$[0,1]$$.) So any of these, basically; you're allowed fractals if you want, as long as they don't intersect themselves and have two definite endpoints. (I'm calling them "arcs" rather than something simple like "curves" is because (a) in mathematics they commonly go under the name Jordan arcs and (b) I want to emphasize that I don't count loops.)

The multiset sum of two sets is like their union, but counting multiplicity. As an example, the multiset sum of a / shape and a \ shape would be a X shape where the center has multiplicity two and all other points have multiplicity one. The multiset sum of > and < could make the same shape.

Show that it is possible for the multiset sum of two arcs to equal the multiset sum of three arcs.

(Note that breaking a line into two pieces won't work because the multiplicity won't be correct where they join.)

• @quarague In fact you can. Homeomorphism is all about continuity. Unlike, say, differentiable functions, continuous functions can have infinite speed. (Checking the definition of homeomorphism, we'll have to show that can define a continuous bijection between the unit interval and the Koch snowflake segment (with bijective inverse). One way to do this is to try to formalize the following function: $f(t)$ equals the point that is $t$ along the Koch segment ($0\le t\le1$).) Dec 15, 2022 at 7:22