Weakly compact cardinal, an infinite cardinal number on which every binary relation has an equally large homogeneous subset
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Limit point compact, a set in which every infinite subset of X has a limit point
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Sequentially compact space, a set in which every infinite sequence has a convergent subsequence
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In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous.