A function is equivariant under the shift map if the transformation on configurations described by commutes with the shift map; that is, for every configuration , it must be the case that . Intuitively, this means that every position of the configuration is updated by using the same rule as every other position. A cellular automaton is defined by a rule for computing the new value of each position in a configuration based only on the values of cells in a prior-fixed finite neighborhood surrounding the position, with all positions of the configuration being updated simultaneously based on the same update rule. That is, the new value of a position is a function only of the values of the cells in its neighborhood rather than depending more generally on an unbounded number of cells of the previous configuration. The function that uses this rule to map a configuration of the cellular automaton into its successor configuration is necessarily equivariant with respect to the shift map, by the assumption that all positions use the same update rule. It is also necessarily continuous in the Cantor topology: if is a fixed pattern, defining a fundamental open set , then is defined by a finite set of patterns, the assignments to cells in the neighborhood of that cause to produce . The Curtis–Hedlund–Lyndon theorem states that these two properties are sufficient to define cellular automata: every continuous equivariant function is the update rule of a cellular automaton. Ceccherini-Silberstein and Coornaert provide the following proof of the Curtis–Hedlund–Lyndon theorem. Suppose is a continuous shift-equivariant function on the shift space. For each configuration , let be the pattern consisting of the single symbol that appears at position zero of . By continuity of , there must exist a finite pattern in such that, if the positions outside are changed arbitrarily but the positions within are fixed to their values in , then the result of applying remains the same at position zero. Equivalently, there must exist a fundamental open set such that belongs to and such that for every configuration in , and have the same value at position zero. These fundamental open sets (for all possible configurations ) form an open cover of the shift space. However, the shift space is a compact space: it is a product of finite topological spaces with the alphabet as their points, so compactness follows from Tychonoff's theorem. By compactness, every open cover has a finite subcover. The finite set of positions appearing in this finite subcover may be used as the neighborhood of position zero in a description of as a cellular automaton rule.