On homologically locally connected spaces

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We provide some properties and characterizations of homologically UVⁿ-maps and lcⁿG-spaces. We show that there is a parallel between recently introduced by Cauty algebraic ANR's and homologically lcⁿG-metric spaces, and this parallel is similar to the parallel between ordinary ANR's and LCⁿ-metric spaces. We also show that there is a similarity between the properties of LCⁿ-spaces and lcⁿG-spaces. Some open questions are raised.