D-6 - Countable Compactness

Abstract

Countable compactness behaves like compactness in some ways. Countable compactness is preserved by continuous images and perfect preimages. Every closed subset of a countably compact space is countably compact; every continuous real-valued function defined on a countably compact space has bounded range. Countable compactness is related to a number of compactness-like properties that evoke countability. Among such properties are sequential compactness (a space is sequentially compact if every sequence has a convergent subsequence), and pseudo compactness (every real valued continuous function is bounded). Countable compactness is equivalent to compactness in many important classes of spaces. Thus, for example, it may be verified that a metrizable space is compact by checking the formally simpler condition of countable compactness. Sequential compactness is the strongest of the three properties, and pseudo compactness is the weakest. Closely related to pseudo compactness is feeble compactness (every locally finite family of open sets is finite). In completely regular spaces, pseudo compactness and feeble compactness are equivalent. Compact Hausdorff spaces are normal (for example, they satisfy the Tietze–Urysohn extension theorem), but countably compact Hausdorff spaces need not be normal.