Oplysninger om | engelsk ord SECOND-COUNTABLE

Eksempler på brug af SECOND-COUNTABLE i en sætning

• from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.
• A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.
• A discrete group is the same thing as a zero-dimensional Lie group (uncountable discrete groups are not second-countable, so authors who require Lie groups to have this property do not regard these groups as Lie groups).
• Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover).
• For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set.
• But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable.
• Note that among Lindelöf spaces, metrizable is equivalent to regular and second-countable, and so the previous result may alternatively be obtained by considering limited information strategies.

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