Inverse limits with subsets of [0,1]×[0,1]

The characterization of chainable continua as inverse limits on [0,1] has resulted in many papers which have contributed to our knowledge of both inverse limits and chainable continua. Quite simple maps on [0,1] give rise to quite complicated inverse limits. In this paper we begin an investigation of continua that can be represented by inverse limits of closed subsets of the unit square I2=[0,1]×[0,1]. We show that some of the usual properties of inverse limits are valid in this situation, give numerous examples, and provide conditions under which the inverse limit is a continuum.