I am trying to find a definition for the open cover of a metric space, but i cannot find it. So, if X is a metric space and A is a subset of X, then what is the definition for open cover of A? Can anyone help? Thank you.
Open cover of a metric space
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0Do you know what an open set (in a metric space) is? – 2012-12-04
2 Answers
An open cover of $A$ in $X$ is simply a family $\mathscr{U}$ of open sets in $X$ such that $A\subseteq\bigcup\mathscr{U}$. A relatively open cover of $A$ as a subspace of $X$ is a family $\mathscr{U}$ of open sets in $A$, i.e., of sets of the form $U\cap A$ for some open $U$ in $X$.
Example: Let $X=\Bbb R$, and let $A=(0,1)$. Then $$\mathscr{U}=\left\{\left(\frac1n,1\right):n\in\Bbb Z^+\right\}$$ is an open cover of $A$, because each $x\in A$ belongs to at least one member of $\mathscr{U}$. Specifically, if $x\in A$, then $x>0$, so there is a positive integer $n$ such that $\frac1n Finally, a set $U\subseteq X$ is open if and only if it is a union of open balls: for each $x\in U$ there is an $\epsilon_x>0$ such that $B(x,\epsilon_x)\subseteq U$, where $B(x,\epsilon_x)=\{y\in X:d(x,y)<\epsilon_x\}$.
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0Thanks, when you say "family of" open sets, do you mean union of open sets? – 2012-12-04
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0@user49065: No, *family*, *collection*, and *set* all mean the same thing: $\mathscr{U}$ is a set of open sets in $X$. I used *family* only because *a set of sets* sounds awkward. – 2012-12-04
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0@user49065: I’ve added a concrete example that may help make it clearer. – 2012-12-04
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0Hi Brian. Did you mean to say $A\subseteq \cup \mathscr{U}$ in the first line? And $X$ instead of $M$? – 2012-12-05
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0@ThomasE.: Yes, and yes; and thanks for catching them. – 2012-12-05
An open cover of a subset $A \subseteq X$, is a collection $\{U_i\}_{i \in I}$ of open sets in $X$, such that
$$A \subseteq \bigcup_{i \in I} U_i$$