Concerning the sum of a countable number of mutually exclusive continua in the plane

In 1918 Sierpiński showed that if the sum of a countably infinite collection of closed point sets is bounded then it is not a continuum. He raised the question weather this theorem remains true if the restriction that the sum should be bounded is removed from the hypothesis. The purpose of the present paper is to show that for the case where each point set of the collection in question is itself a continuum, this question may be answered in the affirmative.