Relating spans of some continua homeomorphic to 𝑆ⁿ

Abstract

Abstract. It has been conjectured that a*(X) > jUq (X) for each nonempty connected metric space X . In this paper we show that o*(X) > (\/3/2)<7q (X) when X C R." is homeomorphic to S"~ for « = 2,3,... and A is convex where A is the bounded component of Rn -X . We also show that under certain conditions a lower bound for the ratio a* (X)la^ (X) is larger than ■\/3/2 . it has also been conjectured that a'(X) > o(X)/2 and that <r0*(X) > aQ(X)/2 for each nonempty connected metric space X . We show that these two inequalities hold when X C Rn is homeomorphic to S"~ ' for « = 3,4,... and A is convex where A is the bounded component of Rn -X .