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- W2014317630 abstract "Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following: D efinition . A topological space X is 1. 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 x 0 ∈ X such that for every neighborhood U of x 0 , | 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]." @default.
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- W2014317630 date "1990-06-01" @default.
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- W2014317630 title "Definitions of compact" @default.
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- W2014317630 doi "https://doi.org/10.2307/2274654" @default.
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