TY - JOUR

T1 - On planarity of compact, locally connected, metric spaces

AU - Richter, R. Bruce

AU - Rooney, Brendan

AU - Thomassen, Carsten

PY - 2011

Y1 - 2011

N2 - Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K 5 or K 3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K 5, K 3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.

AB - Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K 5 or K 3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K 5, K 3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.

U2 - 10.1007/s00493-011-2563-1

DO - 10.1007/s00493-011-2563-1

M3 - Journal article

VL - 31

SP - 365

EP - 376

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 3

ER -