Let $(X,\mathcal{T})$ be a \reftext{def:topological-space-2026a}{topological space}, let $A\subseteq X$, let $I$ be a set, and let $(U_i)_{i\in I}$ be a \reftext{def:family-subfamily-subsets-set-2026a}{family of subsets of $X$}.

We say that $(U_i)_{i\in I}$ is an open cover of $A$ in $X$ if the following two conditions hold.

1. For every $i\in I$, one has $U_i\in\mathcal{T}$.
2. One has

If $(U_i)_{i\in I}$ is an open cover of $A$ in $X$ and if $J\subseteq I$, then the \reftext{def:family-subfamily-subsets-set-2026a}{subfamily} $(U_j)_{j\in J}$ is called a subcover of $A$ if