| Description |
This paper deals with the topological property of compactness and some of the properties of topological spaces which are closely related to compactness. In this paper an attempt is made to trace the development of these concepts from their introduction to their complete acceptance by topologists. Compactness is an extremely strong property for a space to possess. In a compact space every open cover, regardless of its nature, has a finited subcover. It is not immediately obvious that this is an extremely useful and powerful condition. For many years this property was not even associated with the name "compact," and for many more years confusion was caused in articles to the point that it was necessary to define what one meant when employing the term "compact." Other related properties have also had several names applied to their various equivalent conditions. Realcompact and Q-spaces, paracompact and fully normal spaces, hypocompact and S-spaces, almost compact and H-closed spaces are all equivalent pairs, but this was not known at their introduction. Eventually all the properties seem to have come under a name that indicates its relation to compactness by the use of the word compact. In the previously mentioned pairs Q-space, fully normal space, S-space and H-closed space were all the initial terms used to describe the class of spaces in question, but realcompact space, paracompact space, hypocompact space, and almost compact space have replaced them. Throughout this paper it will be the practice to use the term which contains the word compact except where historical accuracy or a proof of equivalency makes it necessary to do otherwise. |
