The Higson compactification $X̅^{d}$ of a non-compact proper metric space (X,d) is rarely equivalent to the Stone-Čech compactification βX. We give a characterization of such spaces. Also, we show that for each non-compact locally compact separable metric space, βX is equivalent to $lim\limits_{⟵ }{X̅^{d}: d$ is a proper metric on X which is compatible with the topology of X}. The approximation method of the above type is illustrated by some examples and applications.
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We provide a necessary and sufficient condition for the Higson compactification to be perfect for the noncompact, locally connected, proper metric spaces. We also discuss perfectness of the Smirnov compactification.
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