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In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' subcover. A strongly Lindelöf space is a topological space such that every open subspace is Lindelöf. Such spaces are also known as hereditarily Lindelöf spaces, because all subspaces of such a space are Lindelöf. Lindelöf spaces are named for the Finnish mathematician Ernst Leonard Lindelöf. == Properties of Lindelöf spaces == In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties, such as paracompactness. But by the Morita theorem, every regular Lindelöf space is paracompact. Any second-countable space is a Lindelöf space, but not conversely. However, the matter is simpler for metric spaces. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf. Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finite products. A Lindelöf space is compact if and only if it is countably compact. Any σ-compact space is Lindelöf. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lindelöf space」の詳細全文を読む スポンサード リンク
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