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Say there is a nested, uncountable sequence of subsets $\{A_i\}_{i \in I}$ of a given set $A$. Can the union of those subsets $\bigcup_{i \in I} {A_i}$ be written as the limit as $i \rightarrow \infty$ of $A_i$?

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\infty to get $\infty$ – Git Gud Jan 20 '13 at 23:31
thanks for the notation – chubbycantorset Jan 20 '13 at 23:31
up vote 3 down vote accepted

Not every uncountable set is $\mathbb R$, not every uncountable set has a natural order, and if we say that the sets are nested it doesn't mean that the index set is linearly ordered, or that it is $\mathbb R$, it could be much much bigger.

The symbol $\infty$ signifies "a point beyond the rest", in the real numbers it signifies that something is getting larger and larger. However outside this context it may not be well-defined. If $I$ is defined in such way that $\infty$ is well-defined as the "supremum" of $I$, then it's fine, but otherwise it might not be a very good practice to use this symbol.

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If the subsets of $A$ are considered objects of a category in which there is a single morphism $A_i \to A_j$ if $A_i \subseteq A_j$, and no morphism otherwise, then for any collection of subsets, $\{A_i\}_{i \in I}$, we have $\cup_{i \in I} A_i = \mathrm{colim}_{i \in I} A_i$. In fact, if the sets $\{A_i\}$ are nested, then this is a directed colimit. So while it may not be entirely precise to write this using the symbols "$i \to \infty$," it's probably not an unforgivable sin to think of it in such terms.

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