# Normal Covering of a Space

I am reading a paper by S. Mardešić and V. Matijević which uses the word "normal covering" of a space $X$ without giving the definition. I am not aware of this term, and I can't really find it in other literature (although to be honest, I haven't made too much of an effort to find it, but I figured I'd ask here anyway.) Is this just another term for an open covering or is there something else to it?

Some context: The authors denote $\text{Cov}(X)$ to be the set of all normal (open) coverings of the space $X$. They use this to define a $\mathcal{U}$-mapping $f: X \rightarrow Y$ which has the property that there exists a normal covering $\mathcal{V} \in \text{Cov}(Y)$ such that $f^{-1}(\mathcal{V})$ refines $\mathcal{U}$, where $\mathcal{U} \in \text{Cov}(X)$.

## 2 Answers

An open cover $U$ of $X$ is normal iff there exists a sequence of open covers $V_n$ such that $V_0=U$ and $V_n$ is a star refinement of $V_{n-1}$.

A covering is normal or regular or Galois if the group of deck transformations act transitively on the fibers. It's called "normal" because this is the case if and only if the fundamental group of the covering space is a normal subgroup of that if the base. See wikipedia for more details.

• Never confuse the concept of a covering map and a cover of a topological space $X$ as a family of subsets $A_\alpha \subset X$ such that $\bigcup_\alpha A_\alpha = X$. Unfortunately notation is not standardized in the literature; some authors use the word covering instead of cover. The question definitely deals with covers. Jul 20, 2018 at 14:41