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Based on the abstract version of the Smital property, we introduce an operator
. We use it to characterise the class of semitopological abelian groups, for which addition is a quasicontinuous operation.
A condition equivalent to sparseness of a set on the plane is formulated and used as a motivation for a new concept of density point on the plane. This is investigated and compared with known previous versions.
The topologies of the density type in Euclidean space of dimension higher than one are introduced. Definitions are based on a notion of density point connected with a set of sequences of real numbers. Our purpose is to study properties of these topologies and connections between them.
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