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Temporal graphs represent interactions between entities over time. Deciding whether entities can reach each other through temporal paths is useful for various applications such as in communication networks and epidemiology. Previous works have studied the scenario in which addition of new interactions can happen at any point in time. A known strategy maintains, incrementally, a Timed Transitive Closure using a dynamic data structure composed of |$O(n^{2})$| binary search trees containing non-nested time intervals. However, space usage for storing these trees grows rapidly as more interactions are inserted. In this paper, we introduce a compact data structure that represents each tree as two dynamic bit-vectors. Furthermore, we present two variants of this data structure: one representing bits in dynamic bit-vectors explicitly and the other representing only the active bits by encoding their consecutive distances. In our experiments, we observed that our data structure improves space usage while having similar time performance for incremental updates when comparing with the previous strategy. The first variant of our data structure gives the best space improvement when constructing Time Transitive Closures for temporally dense temporal graphs, and the second variant uses less space for temporally sparse temporal graphs. [ABSTRACT FROM AUTHOR]