Definition 5

A family of sets $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ is said to be a covering of a set E (or to cover E) if $\mathrm{E} \subset \bigcup\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota$. If $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ and $(\mathrm{Y}_\varkappa)_{\varkappa \in \mathrm{K}}$ are coverings of E, the second of these coverings is said to be finer than the first (or to be a refinement of the first, or to refine the first) if, for each $\varkappa \in \mathrm{K}$, there exists $\iota \in \mathrm{I}$ such that

A set of sets $\mathfrak{R}$ is a covering of E if the family of sets defined by the identity mapping of $\mathfrak{R}$ is a covering of E, in other words, if $\mathrm{E} \subset \bigcup\limits_{\mathrm{X} \in \mathfrak{R}} \mathrm{X}$.

If $\mathfrak{R}$, $\mathfrak{R}'$, $\mathfrak{R}''$ are three coverings of E such that $\mathfrak{R}'$ refines $\mathfrak{R}$ and $\mathfrak{R}''$ refines $\mathfrak{R}'$, it is clear that $\mathfrak{R}''$ refines $\mathfrak{R}$.

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a covering of E. If J is a subset of I such that $(\mathrm{X}_\iota)_{\iota \in \mathrm{J}}$ is still a covering of E, then this covering clearly refines $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$.

¶ Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ and $(\mathrm{Y}_\varkappa)_{\varkappa \in \mathrm{K}}$ be coverings of a set E. Then the family of sets $(\mathrm{X}_\iota \cap \mathrm{Y}_\varkappa)_{(\iota, \varkappa) \in \mathrm{I} \times \mathrm{K}}$ is a covering of E. For if $x \in \mathrm{E}$, there exist indices $\iota \in \mathrm{I}$ and $\varkappa \in \mathrm{K}$ such that $x \in \mathrm{X}_\iota$ and $x \in \mathrm{Y}_\varkappa$, so that $x \in \mathrm{X}_\iota \cap \mathrm{Y}_\varkappa$ Moreover, it is clear that the covering $(\mathrm{X}_\iota \cap \mathrm{Y}_\varkappa)_{(\iota, \varkappa) \in \mathrm{I} \times \mathrm{K}}$ refines each o the coverings $(X_\iota)_{\iota \in I}$, $(Y_\varkappa)_{\varkappa \in K}$. Conversely, let $(Z_\lambda)_{\lambda \in L}$ be a covering of E which refines each of the coverings $(X_\iota)_{\iota \in I}$, $(Y_\varkappa)_{\varkappa \in K}$; if $\lambda \in L$, then there exist indices $\iota \in I$ and $\varkappa \in K$ such that $Z_\lambda \subset X_\iota$ and $Z_\lambda \subset Y_\varkappa$, so that $Z_\lambda \subset X_\iota \cap Y_\varkappa$; hence the covering $(Z_\lambda)_{\lambda \in L}$ is a refinement of

¶ Let $(X_\iota)_{\iota \in I}$ be a covering of a set A, and let $f$ be a mapping of A onto a set B. The family $(f\langle X_\iota \rangle)_{\iota \in I}$ is then a covering of B (Proposition 3), called the image under $f$ of the covering $(X_\iota)_{\iota \in I}$. If $g$ is a mapping of a set C into the set A, the family $(\overset{-1}{g}\langle X_\iota \rangle)_{\iota \in I}$ is a covering of C, called the inverse image under $g$ of the covering $(X_\iota)_{\iota \in I}$. ¶ Let E and F be sets, let $(X_\iota)_{\iota \in I}$ be a covering of E, and let $(Y_\varkappa)_{\varkappa \in K}$ be a covering of F. The family $(X_\iota \times Y_\varkappa)_{(\iota, \varkappa) \in I \times K}$ is then a covering of $E \times F$, called the product of the coverings $(X_\iota)_{\iota \in I}$ of E and $(Y_\varkappa)_{\varkappa \in K}$ of F.