# Cover definition: When is a set a subset of it's cover's union instead of being equal?

In the current Wikipedia entry for covers, it says that

if $${\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace }$$ is an indexed family of subsets $${\displaystyle U_{\alpha }\subset X}$$ (indexed by the set $${\displaystyle A}$$), then $${\displaystyle C}$$ is a cover of $${\displaystyle X}$$ if $${\displaystyle \bigcup _{\alpha \in A}U_{\alpha }\supseteq X}$$.

Since a set cannot contain duplicates and all $$U_\alpha$$ are strict subsets of $$X$$, why does it say $${\displaystyle \bigcup _{\alpha \in A}U_{\alpha }\supseteq X}$$ instead of $${\displaystyle \bigcup _{\alpha \in A}U_{\alpha } = X}$$? What would be an example where $$X$$ is only a subset of its cover's union instead of being equal to it?

• It's a recent edit which lead to this inconsistent formulation. en.m.wikipedia.org/w/… Mar 21 at 0:52

You are right, one could simply require that $$\bigcup_{\alpha \in A} U_\alpha= X$$. However, if you continue reading the Wikipedia article, you will find
Also, if $$Y$$ is a (topological) subspace of $$X$$, then a cover of $$Y$$ is a collection of subsets $$C = \{U_\alpha \}_{\alpha \in A}$$ of $$X$$ whose union contains $$Y$$, i.e., $$C$$ is a cover of $$Y$$ if $$Y ⊆ ⋃_{α ∈ A} U_α .$$ That is, we may cover $$Y$$ with either sets in $$Y$$ itself or sets in the parent space $$X$$.
I think this is the reason for requiring $$\bigcup_{\alpha \in A} U_\alpha \supseteq X$$ - it keeps notation consistent in both situations.
Both are equivalent, as we always have $$\displaystyle\bigcup_{\alpha\in A}U_\alpha\subseteq A$$. Nonetheless, one usually writes the condition as $$\displaystyle\bigcup_{\alpha\in A}U_\alpha\supseteq A$$, as that reinforces the idea of a cover: a collection of sets whose union encloses all of $$A$$.
• In fact the question could rather be, "Why is the condition $U_\alpha \subset X$ included in the definition of "cover?". Does it serve any useful purpose? Mar 20 at 18:54