Inverse limits with subsets of \([0,1]\times[0,1]\)
If we investigate inverse limits obtained by means of sequences of maps of the closed interval into itself, \( I_1 \leftarrow I_2 \leftarrow \dots \) with the bonding maps \( f_n:I_{n+1} \rightarrow I_n \), we can regard them as subspaces of the Hilbert cube \( I_1 \times I_2 \times \dots\), consisting of points \( (x_1, x_2,\dots) \) with conditions in the form of equations \( x_n = f_n (x_{n+1}) \) imposed on their consecutive coordinates. In other words, the points \( (x_{n+1}, x_n) \) must lie on the graphs of corresponding maps \( f_n \). The author proposes a generalization of this condition assuming only that the points in question should lie on subsets \( M_n \) of the squares \( I_{n+1} \times I_n \) that look like graphs of maps, projecting, for instance, onto the \( x \)-axis. Even the special case when the sets \( M_n \) are equal, and closed (this condition assures the compactness of the generalized limit) is of interest, generalizing situations known from the practice with inverse limits. The generalized limit is not necessarily chainable. For instance, the limit associated with the set \( M \) which is the union of the diagonals of the square, is topologically the Cantor fan. In the examples known to the author, the dimension of the generalized limit remains \( 1 \), unless it is infinite, and this is a conjecture stated at the end of the paper.
scientific article; zbMATH DE number 2093562