ABSTRACT
The term \compact" was initially coined to describe the property of a metric space in which every innite subset has a limit point. However, this sense of the term failed to give some desirable theorems for topological spaces, especially its invariance under the product topology. On the other hand, it was found that, in metric spaces, this property is equivalent to the Borel{Lebesgue property: Each open cover has a nite subcover. After Tychono proved the invariance of this last property under the formation of topological products, the following denition of compactness was universally adopted.
