We prove that the Almgren–Pitts 6-width of unit 3-ball is less than $$2\pi $$ . also there exists a free boundary minimal surface in has genus at most 1, index 5, area , and not equatorial disk or critical catenoid.

We study minimal sets on continua X with a dense free interval J and locally connected remainder. This class of includes important spaces such as the topologist's sine curve or Warsaw circle. In case when remainder are known is connected, we obtain full characterization topological structure sets. particular, given in X∖J local dendrite.