Abstract
In Igusa and Todorov(2013) we constructed topological triangulated categories as stable categories of certain topological Frobenius categories . In this paper we show that these categories have a cluster structure for certain values of c including c = pi. The continuous cluster categories are those which have cluster structure. We study the basic structure of these cluster categories and we show that is isomorphic to an orbit category of the continuous derived category if c = r pi/s. In , a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.