First consider the following definitions from Baby Rudin:(adsbygoogle = window.adsbygoogle || []).push({});

Interval:A set of real numbers of the form [itex][a,b][/itex] where for all [itex]x \in [a,b][/itex] we have [itex]a \le x \le b[/itex].

K-Cell:A set of k-dimensional vectors of the form [itex]x = (x_1, ...,x_k)[/itex] where for each [itex]x_j[/itex] we have [itex]a_j \le x_j \le b_j[/itex] for each [itex]j[/itex] from [itex]1 \le j \le k[/itex].

Clearly, a one-dimensional k-cell is an interval. But I'm confused about the relationship between a multi-dimensional k-cell and an interval. For example, in Theorem 2.40, Rudin speaks of "subdividing" some k-cell [itex]I[/itex] into smaller intervals [itex]Q_i[/itex], such that the union of [itex]Q_i[/itex] is precisely [itex]I[/itex].

So suppose we are dealing with a multi-dimensional k-cell (i.e., let [itex]I[/itex] be a k-cell with [itex]k > 1[/itex]). And suppose further we fix [itex]c_j = (a_j + b_j)/2[/itex] to then construct the two intervals [itex][a_j,c_j][/itex] and [itex][c_j,b_j][/itex]. According to Rudin, we have then just created [itex]2^k[/itex] k-cells named [itex]Q_i[/itex]. But it seems to me that since the k-cells of [itex]Q_i[/itex] are precisely intervals, that the union of these intervals could not possibly equal [itex]I[/itex] since the union of intervals must be another interval which [itex]I[/itex] is not.

What am I missing? How does the union of [itex]Q_i[/itex] equal [itex]I[/itex] in this case.

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# Relationship Between K-Cells and Intervals in Baby Rudin

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