More popular in the 19th and early 20th centuries was the Bolzano Weierstrass criterion that every sequence admits a convergent subsequence, now called sequential compactness.
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Sequential compactness of " B " in this metric can be shown by a diagonalization argument similar to the one employed in the proof of the Arzel? Ascoli theorem.
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So it is natural to hope that a suitable notion of convergence in arbitrary spaces will lead to a compactness criterion generalizing sequential compactness in metrizable spaces that will be as easily applied to deduce the compactness of products.
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These metric spaces have some nice properties like : in a metric space compactness, sequential compactness and countable compactness are equivalent etc . These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in \ scriptstyle \ mathbb R.
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