Abstract
In the present work, we introduce the notion of pseudo-seminorm, then we established, a criterion for the metrizability of a pseudo vector space 𝛯 with a pseudo topology 𝜏. It is specified, that for the metrizability of 𝛯 is necessary and sufficient that 𝛯 is first countable. Moreover, a sufficient condition for existence a base for the filter of neighborhoods of zero is proved and demonstrated that this condition introduces a compatible pseudo topology with the algebraic vector space structures.
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