Definitions of compact

Abstract

Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following:Definition. A topological space X is1. Compact(1) if every open cover of X has a finite subcover.2. Compact(2) if every infinite subset E of X has a complete accumulation point (i.e., a point x0 ∈ X such that for every neighborhood U of x0, |E ∩ U| = |E|).3. Compact(3) if there is a subbase S for the topology on X such that every cover of X by members of S has a finite subcover.4. Compact(4) if each nest of closed, nonempty sets has a nonempty intersection.5. Compact(5) if every family of closed sets in X which has the finite intersection property (every finite subfamily has a nonempty intersection) has a nonempty intersection.6. Compact(6) if each net in X has a cluster point.7. Compact(7) if each net in X has a convergent subnet.This work was motivated primarily by consideration of various proofs that the Tychonoff theorem, T (“the product of compact topological spaces is compact”) is equivalent to the Axiom of Choice, AC. Tychonoff's original proof that AC implies T used Definition 2 [13]. Other proofs have used Definitions 3 and 5; see [5]. The proof by Kelley that T implies AC uses Definition 5 [6].