Some universally pierced arcs in 𝐸³

Loveland, L. D.

doi:10.1090/S0002-9939-1975-0370590-1

Some universally pierced arcs in $E^{3}$
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Proc. Amer. Math. Soc. 49 (1975), 469-474 Request permission

Abstract:

A subset $X$ of ${E^3}$ is said to be universally pierced if each $2$-sphere containing $X$ can be pierced by a tame arc at each point of $X$. We show that an arc $A$ is universally pierced provided $A$ has a shrinking point $p$ such that either $p$ lies in a tame arc in $A$ or ${E^3} - A$ has $1$-ALG at $p$. Applying this result we show the existence of infinitely many wild universally pierced arcs.

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