Paracompact Hausdorff space

Definition
A topological space is termed paracompact Hausdorff if it satisfies the following equivalent conditions:


 * 1) It is paracompact and Hausdorff
 * 2) Given any open cover of the space, there is a partition of unity subordinate to that open cover; in other words, there is a partition of unity such that the support of each function is contained in some set of that open cover
 * 3) It is regular and every open cover has a locally finite open refinement
 * 4) It is regular and every open cover has a locally finite closed refinement
 * 5) It is regular and every open cover has a locally finite refinement
 * 6) It is regular and every open cover has a countably locally finite open refinement

The second definition is the one used in algebraic topology.