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Keywords:
Barrelled space; convex-Baire space; normed lattice; pairwise Baire spaces; quasi-Baire spaces; quasi-uniformity
Barrelled space; convex-Baire space; normed lattice; pairwise Baire spaces; quasi-Baire spaces; quasi-uniformity
Summary:
We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that $E$ is a quasi-Baire space iff $(E, T({\mathcal U}),T({\mathcal U}^{-1}))$, is a pairwise Baire bitopological space, where $\mathcal U$, is a quasi-uniformity that determines, in $L$. Nachbin’s sense, the topological ordered space $E$.
References:
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[2] Ferrer, J; Gregori, V; Alegre, C.: Quasi-uniform structures in linear lattices. Rocky Mountain J. Math. (1994), 877–884. MR 1245452
[3] Fletcher, P.; Lindgren, W.F.: Quasi-uniform Spaces. Marcel Dekker Inc. New York, 1982. MR 0660063
[4] Kelly, J,C.: Bitopological spaces. Proc. London Math. Soc. (3) 13 (1963), 71–89. MR 0143169 | Zbl 0107.16401
[5] Nachbin, L.: Topology and Order. Robert E. Kriegler Publishing Co., Huntington, New York, 1976. MR 0415582 | Zbl 0333.54002
[6] Valdivia, M.: Topics in Locally Convex Spaces. North-Holland, Amsterdam, 1982. MR 0671092 | Zbl 0489.46001
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