A question about Heine-Borel Theorem.

Question

Im reading chapter2 of rudin's Principle of Mathematical analysis. Heine-Borel theorem is involved in this chapter,

$\mathbf{2.41}\,\,$ Theorem$\,\,\,$ If a set $E$ in $R^k$ has one of the following three properties, then it has the other two:

$\quad\text{(a)}\,\,$ $E$ is closed and bounded.

$\quad\text{(b)}\,\,$ $E$ is compact.

$\quad\text{(c)}\,\,$ Every infinite subset of $E$ has a limit point in $E$.

I think that the k in $R^k$ should be finite, is that right?

Usually $k$ stands for a nonnegative integer when placed in "$\Bbb R^k$". — Pedro, Jan 03 '14 at 22:33
For generalizations, go to http://math.stackexchange.com/questions/104777/are-there-more-general-spaces-than-euclidean-spaces-to-have-the-heineborel-prop — ir7, Jan 03 '14 at 22:35

score 0 · Accepted Answer · answered Jan 26 '15 at 15:45

0

Yes, k means a positive integer. FYI, in Rudin’s POMA, the symbol $R^k$ first appears in p.16.

answered Jan 26 '15 at 15:45

Eli4ph

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A question about Heine-Borel Theorem.

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