A subset B₀ of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$  of closed balanced and bornivorous subsets of X such that B_i+1 + B_i+1 ⊆ B_i for all i = 0, 1, .... In this case, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$  is called a defining sequence for B₀. A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.^[1]

A locally convex quasi-ultrabarrelled space is quasi-barrelled.^[1]

Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.^[1]

• Barrelled space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Ultrabarrelled space
• Uniform boundedness principle#Generalisations

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