Question

Can Nice Spaces be Broken into Two Equal Parts? Stephen Watson York University

Answered in Can Nice Spaces be Broken into Two Equal Parts? by Attilio Le Donne
Answered in Can Nice Spaces be Broken into Two Equal Parts? by Mikhail Matveev

These questions below are motivated by results of Kouki Taniyama who proved in his preprint ``Dividing a topological space into mutually disjoint and mutually homeomorphic space'' that both the reals and [0,1] and, in fact, any separable locally finite one-dimensional simplicial complex can be so divided. This result was presented at the 1999 Yokohama topology conference.
1. Can every complete (Borel) metric space without isolated points be partitioned into two homeomorphic subspaces?

2. Is there a subset of [0,1] without isolated points which cannot be partitioned into two homeomorphic subsets?

Related literature includes W.Gustin in Ann. Math. 54 (1951) 250-261. and B. Halpern in Amer. Math. Monthly 75(1968) 1073-1077.

Received by the editors: January 14, 2000