Might mean (unverified): Space where every barrelled subspace.
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We found one dictionary that defines the word quasibarrelled:

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  1. quasibarrelled: Wiktionary

Definitions from Wiktionary (quasibarrelled)

adjective:  (mathematics, of a topological vector space) For which every barrelled bornivorous set in the space is a neighbourhood of zero.

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