{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T23:13:40Z","timestamp":1723245220820},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,6]]},"abstract":"<jats:p>Several definitions of \u201ccompact\u201d for topological spaces have appeared in the literature (see [5]). We will consider the following:<\/jats:p><jats:p>D<jats:sc>efinition<\/jats:sc>. A topological space <jats:italic>X<\/jats:italic> is<\/jats:p><jats:p>1. <jats:italic>Compact<\/jats:italic>(1) if every open cover of <jats:italic>X<\/jats:italic> has a finite subcover.<\/jats:p><jats:p>2. <jats:italic>Compact<\/jats:italic>(2) if every infinite subset <jats:italic>E<\/jats:italic> of <jats:italic>X<\/jats:italic> has a complete accumulation point (i.e., a point <jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub> \u2208 <jats:italic>X<\/jats:italic> such that for every neighborhood <jats:italic>U<\/jats:italic> of <jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, |<jats:italic>E<\/jats:italic> \u2229 <jats:italic>U<\/jats:italic>| = |<jats:italic>E<\/jats:italic>|).<\/jats:p><jats:p>3. <jats:italic>Compact<\/jats:italic>(3) if there is a subbase <jats:italic>S<\/jats:italic> for the topology on <jats:italic>X<\/jats:italic> such that every cover of <jats:italic>X<\/jats:italic> by members of <jats:italic>S<\/jats:italic> has a finite subcover.<\/jats:p><jats:p>4. <jats:italic>Compact<\/jats:italic>(4) if each nest of closed, nonempty sets has a nonempty intersection.<\/jats:p><jats:p>5. <jats:italic>Compact<\/jats:italic>(5) if every family of closed sets in <jats:italic>X<\/jats:italic> which has the finite intersection property (every finite subfamily has a nonempty intersection) has a nonempty intersection.<\/jats:p><jats:p>6. <jats:italic>Compact<\/jats:italic>(6) if each net in <jats:italic>X<\/jats:italic> has a cluster point.<\/jats:p><jats:p>7. <jats:italic>Compact<\/jats:italic>(7) if each net in <jats:italic>X<\/jats:italic> has a convergent subnet.<\/jats:p><jats:p>This work was motivated primarily by consideration of various proofs that the Tychonoff theorem, T (\u201cthe product of compact topological spaces is compact\u201d) 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].<\/jats:p>","DOI":"10.2307\/2274654","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:35:29Z","timestamp":1146940529000},"page":"645-655","source":"Crossref","is-referenced-by-count":7,"title":["Definitions of compact"],"prefix":"10.1017","volume":"55","author":[{"given":"Paul E.","family":"Howard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026049_ref011","volume-title":"Equivalents of the axiom of choice. 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