We propose point with implicit connectivity (PIC) as a new data structure for representing solid objects using points. In the PIC representation, an object is adaptively sampled into an octree, where each leaf cell contains at most one surface component of the object. Each surface component is represented by a vertex, together with inside/outside classification values of the cell corners. PIC objects are compact, feature-preserving, and supports easy construction of the boundary surfaces. To convert geometric objects into the PIC representation, we propose a sampling algorithm and use quadric error functions as error metrics. For CSG operations between PIC objects, we present a feature-preserving, adaptive CSG algorithm on the octrees. Our experiments show promising results for PIC objects with sharp features and large flat regions.