Erdősův prostor

Barna, Mikuláš

This thesis explores the topological properties and characterizations of Erdős space (E) and complete Erdős space (Ec). Originally introduced by Paul Erdős in 1940 as examples of totally disconnected yet one-dimensional spaces, these sets serve as the archetypal models for the class of almost zero-dimensional spaces. We establish the fundamental properties of almost zero-dimensional spaces proving they are hereditary, productive, totally disconnected and at most one dimensional. A key topological characterization is presented, identifying these spaces as the set of endpoints of an R-tree. We then focus on structures of E and Ec. By introducing the Lelek function and the Lelek fan, the author demonstrates that complete Erdős space can be represented as a dense set of endpoints in a smooth fan. The thesis concludes with a core characterization theorem, proving that every almost zero-dimensional space can be embedded into both Erdős and complete Erdős space, and can be represented as the graph of a semi-continuous function over a zero-dimensional domain.

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