© 1997 Copyright by Sonia Sabogal. All rights reserved.

Question

Self-similarity Sonia Sabogal Answered in S1 is a self-similar symbolic space by F. G. Arenas and M. A. Sánchez-Granero

1. A quotient X of Cantor's space S^N is called a self-similar symbolic space (see Kameyama A., Japan J. Indust. Appl. Math. 10(1993), 85-95) if X is homeomorphic to S^N/R such that the equivalence relation R satisfies: xRy implies ixRiy for all i in S, x,y in S^N.
   Question: Is the circle S¹ a self-similar symbolic space?

2. A topological space X is called self-similar (topologically) if every nonempty open set contains a subspace homeomorphic to X.

   1. Under what conditions is a self-similar space a self-similar symbolic space?

   2. Is every self-similar space homeomorphic to a product of the form X^Y, with Y an infinite set?

   3. Does Xⁿ self-similar imply X self-similar?

3. Is the set N of natural numbers with the cofinite (finite complement) topology a quotient of Cantor's space?

References

1. W. J. Charatonik and A. Dilks, Self-Similarity, Topol. and its Appl.; 55 (1994), 215-238.