Abstract
A directed set is a non-empty set S equipped with a reflexive and transitive relation ≤ such that for any r, s ∈ S, there is t ∈ S with r ≤ t and s ≤ t. An inverse system \((X_s, f^s_r; S)\) of spaces consists of a directed set S, a space X s for each s ∈ S and bonding maps \(f^s_r: X_s\to X_r\), for r, s ∈ S with r ≤ s, such that \(f^s_s\) is the identity on X s and \(f^t_r=f^s_r \circ f^t_s\) whenever r ≤ s ≤ t. Evidently, the equality \(f^t_r=f^s_r \circ f^t_s\) need only be checked for r < s < t.
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Charalambous, M.G. (2019). Inverse Limits and N-Compact Spaces. In: Dimension Theory. Atlantis Studies in Mathematics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-22232-1_16
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