totally paracompact

Definition 1

A space is said to be totally paracompact if every open base has a locally finite subfamily covering the space.

Definition 2

Let X be a topological space and let Y be its subset.
Y is called totally paracompact with respect to X if every outer base contains a X-locally finite cover. A family B of open sets in X is called an outer cover if, for every point y in Y and every open set G in X, there is some member V of B such that x is contained in V and that V is contained in G.

Remark

• For Definition 1, it is also called basically paracompact

Reference

Definition 1

R. M. Ford, Basic properties in dimension theory, Dissertation, Auburn University, 1963.
M. K. Singal, Some Generalizations of Paracompactness, Proceedings of the Kanpur topological conference, 1968. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1971. pp. 245-263.

Definition 2

R. Telgarsky and H. Kok, The space of rationals is not absolutely paracompact, Fund. Math. 73 (1971), 75-78.