The theory of distributed computing shares a deep and fascinating connection with combinatorial and algebraic topology. One of the key ideas that facilitates the development of the topological theory of distributed computing is the use of iterated shared memory models. In such a model processes communicate through a sequence of shared objects. Processes access the sequence of objects, one-by-one, in the same order and asynchronously. Each process accesses each shared object only once. In the

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... t basic form of an iterated model, any number of processes can crash, and the shared objects are snapshot objects. A process can write a value to such an object, and gets back a snapshot of its contents. The purpose of this paper is to give an introduction to this research area, using an iterated model based on the safe-consensus task (Afek, Gafni and Lieber, DISC'09). In a safe-consensus task, the validity condition of consensus is weakened as follows. If the first process to invoke an object solving a safe-consensus task returns before any other process invokes it, then the process gets back its own input; otherwise the value returned by the task can be arbitrary. As with consensus, the agreement requirement is that always the same value is returned to all processes. A safe-consensus-based iterated model is described in detail. It is explained how its runs can be described with simplicial complexes. The usefulness of the iterated memory model for the topological theory of distributed computing is exhibited by presenting some new results (with very clean and well structured proofs) about the solvability of the (n, k)-set agreement task. Throughout the paper, the main ideas are explained with figures and intuitive examples.