What is the topology of a phase diagram?
Particularly since Fe was mentioned, I'll focus there. For pure iron, it is well known that $\alpha$-Fe, the stable phase at room temperature, is bcc. This is stable up to 1184K where it transitions to fcc $\gamma$-Fe. In turn, this transforms back to bcc $\delta$-Fe at 1667K before melting at 1810K. Yet, both $\alpha$ and $\delta$ have a bcc crystal structure. 

If one looks at the Fe-Zn phase diagram:

one sees that the region of bcc phase, enclosed in dark blue, fully encompasses the green sliver of fcc $\gamma$. Using the Zn concentration as a free parameter, one can go from $\alpha$ to $\delta$ without changing phase. 

In contrast, in the Fe-Pd phase diagram:

one sees a different picture. Here, Fe and Pd show complete miscibility in the fcc phase across the entire diagram. This means you can't go from $\alpha$ to $\delta$ bcc phase without going through the fcc phase. Yet, one can see that, by trading off Zn for Pd, one could manage to get around the fcc phase by some path in composition space.