weak cover compact

Definition

A topological space X is weak cover compact if every open cover of X has a weak cover compact open refinement. Here, a "weak cover compact cover" is defined as following. Let V be a cover of X and assume:

1. V_i is a uncountably infinite collection which consists of distinct members in V;
2. p_i and q_i are points in V_i without repitition;
3. a net {p_i} has a limit point.
   If the above conditions impliy the existense of a limit point of {q_i}, we call V weak cover compact.

Remark

• We will use the abbreviation wcc for weak cover compact.

Reference

• Mancuso, V. J.,Mesocompactness and related properties, Pacific J. Math. 33 1970 345--355.