Cover (Topology)

Definition

A cover (or covering) of a topological space $X$ is a family of subsets $\{U_i\subseteq X{\}}_{i\in I}$ such that $X={\bigcup }_{i\in I}{U}_i.$ Often one means an open cover, where each $U_i$ is open in $X$.

In categorical or sheaf-theoretic contexts, one sometimes speaks of a family of maps $\{U_i\to X{\}}_{i\in I}$ as a cover if the images jointly cover $X$ (and possibly satisfy additional locality/gluing axioms). This notion is still a cover by subsets (via the images), not a covering space.

Covering space vs cover

A covering space of $X$ is a continuous map $p:E\to X$ such that every point $x\in X$ has an open neighbourhood $U$ for which $p^{-1}(U)\cong {\coprod }_{j\in J}U,$ and the restriction $p^{-1}(U)\to U$ is a disjoint union of homeomorphisms $U\to U$ (one per component). Intuitively, a covering space looks locally like “many disjoint copies” of the base.

By contrast, a (open) cover of $X$ is just a family of subsets whose union is $X$.

• cover of $X$ = family of subsets (often open) whose union is $X$.
• covering space of $X$ = a single space $E$ mapping to $X$ that locally looks like many disjoint copies of $X$.

Examples

• The projection $\mathbb{R}\to S^1$, $t\mapsto e^{2\pi it}$, is a covering space of the circle. Small arcs in $S^1$ lift to disjoint intervals in $\mathbb{R}$.
• Two overlapping open arcs that union to $S^1$ form an open cover of $S^1$, but they are not a covering space of $S^1$.

Related Concepts

Covering Space Fibre Bundle Topological Fibration Hurewicz Fibration Serre Fibration