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Let $X$ be a topological space,

Ive seen some define a cover, as a collection, $A$, of sets such that $X=\bigcup A$, while others use the condition $X\subseteq \bigcup A$.

Why the difference? My lecturer uses the first definition but to me the second seems more natural. After all, to cover something doesn't necessarily mean to cover it exactly.

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    $\begingroup$ In the places where you've seen this, is each of the sets $A$ a subset of $X$? If so, the first definition is appropriate, because $X = \bigcup A$ if and only if $X \subseteq \bigcup A$. If not, then the second definition is appropriate. $\endgroup$ – Lee Mosher Apr 8 '16 at 15:40
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    $\begingroup$ For example, if you were asking for a countable open cover of $\mathbb{R}$, and you considered open intervals, say $B=\{(-n,n)\}$, then it would be technically correct to say $\mathbb{R} \subseteq B$, but that's not everything you can say, and also feels strange. It depends on the context, so if it is an open cover where each $A_i \subset X$, then $X=\bigcup A_i$ $\endgroup$ – Andres Mejia Apr 8 '16 at 15:44
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The difference is : is $X$ a subspace of a larger space $Y$ ?

If it isn't, you only have one natural definition : a cover is a collection of subspaces $A_i$ such that $X = \bigcup_i A_i$.

But if $X\subset Y$, then you have two natural definitions : a cover of $X$ can either be a collection of subspaces of $X$ such that $X = \bigcup_i A_i$ (def $1$), or a collection of subspaces of $Y$ such that $X\subset \bigcup_i A_i$ (def $2$).

Of course a cover for def $1$ is in particular a cover for def $2$. And if you have a cover $(A_i)$ for def $2$, you get a cover for def $1$ by using $B_i = A_i\cap X$.

This behaves well because for instance $B_i$ is open/closed in $X$ iff $A_i$ is open/closed in $Y$. On the other hand, if $(A_i)$ is an open cover for def $1$, it may not be an open cover for def $2$, because an open subset of $X$ is not necessarily open in $Y$ if $X$ is not itself open if $Y$. So in that case you have to replace $A_i$ with $A'_i\subset Y$ such that $A'_i$ is open in $Y$ and $A_i = X\cap A'_i$ (which exists by definition).

In the end you can see that the two definitions give the same thing if you're ready to modify a little your covers when you switch definitions.

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  • $\begingroup$ What you've said is fine if $X$ is an open subset of $Y$. Otherwise either a cover for def 1 is not a cover for def 2. If $X$ is a general subset of $Y$ and you want to use definition 1, you should take the $A_i$ to be open subsets of $X$ in the relative topology. $\endgroup$ – Aidan Sims Apr 8 '16 at 15:26
  • $\begingroup$ It is always fine as I wrote it because the question is not about "open covers" but simply about "covers". But it's true that an open cover for definition $1$ will not always be an open cover for definition $2$ if $X$ is not open in $Y$, I'll edit to take that into account. $\endgroup$ – Captain Lama Apr 8 '16 at 15:33
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$X\subset\bigcup A\implies X=\bigcup \left(X\cap A\right)$

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