Tree of shapes

Theorem: if $X$ is unicoherent and locally connected, $u$upper semicontinuous,
                    then shapes of $u$ are disjoint or nested

Examples: $X=\mathbb {R}^n$, $X=$ Jordan domain, $X=\mathbb {S}^n$ ($n\geq2$)...

Counter-examples: $X=\mathbb {S}^1$, $X=$ cylinder, $X=$ torus...

$u$ not semicontinuous:

Summer School on Mathematical Problems in Image Processing, Trieste, Italy, September 4th-18th 2000