Whitney Map

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Let $(X,\tau)$ be a compact metric space. Let $\mathcal{H} \subset 2^X $. A Whitney map for $\mathcal{H}$ is a continuous function $\mu \colon \mathcal{H} \rightarrow [0,\infty)$ such that

  1. for any $A,B \in \mathcal{H}$ with $A \subset B$ and $A \neq B$, we have $\mu(A) < \mu(B)$;
  2. $\mu(A)=0$ if and only if $A \in \mathcal{H} \bigcap \mathcal{F}_1(X)$.


All compacta admit a Whitney map
Being a pseudo-arc is a Whitney property
Being a pseudo-arc is a sequential strong Whitney-reversible property