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Let $[a, b]$ be a given closed, bounded interval and let $X$ be a subset of $[a,b]$. A collection $\textbf{C}$ of closed subintervals of $[a, b]$ is a full cover of $X$ if to each $x$ in $X$ there corresponds a number $\delta(x) > 0$ such that every closed subinterval of [a, b] that contains $x$ and has length less than $\delta(x)$ belongs to $\textbf{C}$.

My comments: why do we want arbitrarily small closed intervals about pts of our subset $X$? What does this do for us? Is this just another way of saying take $I \subset [a,b]$ and this notion of full cover will create a closed interval $[c,d]$ where $I\subset [c,d] \subset[a,b]$? But what if $I$ is already closed? Will this just cover it again? Does this full cover simply mean the smalles closed set containing any subset of out closed and bounded interval $[a,b]$?

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Full covers are useful in that assuming them is the assumption of Cousin's Theorem, whose conclusion is a partition of $[a,b]$ into finitely many closed subintervals each in the full cover. See en.wikipedia.org/wiki/Cousin's_theorem –  coffeemath Mar 6 '14 at 8:58

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