Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $A_i$?

share|improve this question
1  
\infty to get $\infty$ –  Git Gud Jan 20 '13 at 23:31
    
thanks for the notation –  chubbycantorset Jan 20 '13 at 23:31

2 Answers 2

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.

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.