Udayan B. Darji

Covering ℝ with translates of a compact set

Motivated by a question of Gruenhage, we investigate when R is the union of less than continuum many translates of a compact set C C R. It will follow from one of our general results that if a compact set C has packing dimension less than 1, then R is not the union of less than continuum many translates of C.

Classification problems in continuum theory

We study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics. We provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes. For dendrites with finitely many branch points the homeomorphism and quasi-homeomorphism classes coincide, and the minimal quasi-homeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the homeomorphism relation between dendrites is

Complexity of hereditarily decomposable continua

S_{\infty }

Approximation of automorphisms of the rationals and the random graph

-universal. It is shown that the classes of trees and graphs are both

UNIVERSALLY FIRST RETURN CONTINUOUS FUNCTIONS

D_2( \boldsymbol \Sigma^0_3)

Visible and Invisible Cantor Sets

-complete, the class of dendrites is

On the Existence of

\boldsymbol \Pi^0_3

C^1

-complete, and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is

Functions with Perfect Level Sets

\boldsymbol \Pi^0_3

Two General Extension Theorems

-complete. We also show that if