nLab fully normal topological space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A topological space is called fully normal if every open cover {U iX} iI\{U_i \subset X\}_{i \in I} is a normal cover, i.e., has a refinement by an open cover {V jX} jJ\{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} jJ\{V_j\}_{j\in J} also admits such a star refinement, and this process can be continued indefinitely.

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

(1)star(x,𝒱){V j𝒱|xV J} 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.



Last revised on June 16, 2021 at 08:04:33. See the history of this page for a list of all contributions to it.