Approximative compactness of the algebraic sum of sets

A. I. Vasil'ev

Institute of Mathematics and Mechanics, Ural Scientific Center of the AS of USSR

Abstract: Let $X$ be a group with an invariant metric, $A$ and $B$ nonempty subsets of $X$ with $B$ compact. It is proved that if $A$ is an existence set [1] (approximatively compact [2]) then $A+B$ and $B+A$ are existence sets (approximatively compact). An example is given of a one-dimensional linear metric space (with an invariant metric) in which there exist an approximatively compact set $A$ and an element $v$ such that $A+v$ is not an existence set.

UDC: 517.5

Received: 19.03.1976

 English version: Mathematical Notes, 1978, 23:1, 32–34

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