meet, all of them either time-oriented away from the vertex (source) or toward it (sink), as illustrated by Fig. \[fig:sourcesink2d\]. For compact spatial boundary conditions, choosing a source and a sink as initial and final (degenerate spatial) boundaries will convert the cylinder into a spherical $S^2$-spacetime topology. A similar choice of boundary conditions in 2+1 dimensions will lead to a $S^3$-spacetime topology, with the source and sink forming the south and north pole of the sphere, as we will see later.

There is a particular substructure of the triangulations, called a [*bubble*]{} [@hoekzema], which involves the newly added building blocks and is helpful in analyzing the geometry of nonfoliated CDT. In 1+1 dimensions it is simply a pair of $sst$-triangles with a chain of $tts$-triangles in between. This is the general structure of a two-dimensional connected region bounded by a closed loop of spacelike links, and whose interior contains only timelike links, as shown in Fig. \[fig:bubblestrip\], left (we are assuming that vertex causality is satisfied everywhere). This should be contrasted with the structure of a strip, which likewise denotes a two-dimensional piece of triangulations bounded by spacelike links and without spacelike links in its interior, but whose boundary is 