nLab fully normal topological space

Definition

A topological space is called fully normal if every open cover $\{U_i \subset X\}_{i \in I}$ is a normal cover, i.e., has a refinement by an open cover $\{V_j \subset X\}_{j \in J}$ such that every star (1) in the latter cover is contained in a patch of the former. Furthermore, the resulting cover $\{V_j\}_{j\in J}$ also admits such a star refinement, and this process can be continued indefinitely.

Here, for $x \in X$ a point, the star of $x$ is the union of the patches that contain $x$:

(1)$$star(x,\mathcal{V}) \;\coloneqq\; \left\{ V_j \in \mathcal{V} \;\vert\; x \in V_J \right\}$$

Normal covers are also known as numerable covers, since they are precisely the open covers that admit a subordinate partition of unity.

In pointfree topology

Any completely regular locale has a largest uniformity, the fine uniformity, which consists of all normal covers.

If a completely regular locale admits a complete uniformity, then the fine uniformity is complete.

A locale is paracompact if and only if it admits a complete uniformity. In this case, we can take the fine uniformity.

Examples

• metric spaces are fully normal

Properties

• fully normal spaces are equivalently paracompact

Related conceps

• numerable cover

• paracompact space

• paracompact locale

• separation axiom