Sequential strong Whitney-reversible property

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Let $P$ be a topological property. Then $P$ is said to be a sequential strong Whitney-reversible property provided that whenever $X$ is a continuum such that there is a Whitney map $\mu$ for $C(X)$ as a sequence $\{t_n\}_{n=1}^{\infty}$ in $(0,\mu(X))$ such that $\lim t_n=0$ and $\mu^{-1}(t_n)$ has property $P$ for each $n$, then $X$ has property $P$.