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We consider digits-deleted sets or Cantor-type sets with $\beta$-expansions.We calculate the Hausdorff dimension $d$ of these sets and show that $d$ iscontinuous with respect to $\beta$. The $d$-dimentional Hausdorff measure ofthese sets is finite and positive.

Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets