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FACT useful in these problems: The union of a countable collection of countable sets is countable. (a) Prove that for any n ∈ N, the collection of subsets of N having cardinality n is countable.

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Students preparing for courses in real analysis often encounter either very exacting theoretical treatments or books without enough rigor to stimulate an in-depth understanding of the subject.

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Did you know that some nouns in English can be both countable AND uncountable, depending on the situation? It's true! For these words, the uncountable form usually refers to the general idea, and the countable form usually refers to a specific item.

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Money is countable, nothing but some numbers. El dinero es contable, Son nada más que números. A set in which the number of elements is limited, countable. Conjunto en el que el número de elementos es limitado, contable. A Hilbert space is separable if and only if it admits a countable orthonormal basis.

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a, some, any - countable and uncountable nouns. A car, some money, any money. Some nouns are uncountable in English, but they are countable in other languages. Some of them are: advice, news (it ends in -s, but it's a singular word), furniture, luggage, baggage, bread, cheese, toast, etc.

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Sep 19, 2013 · open if it can be represented as a union of open intervals. subsets of S which contains ˘ and S and is closed under ﬁnite inter-sections and arbitrary (countable or uncountable!) unions. The ele-ments of t are often called the open sets. A set S on which a topology is chosen (i.e., a pair (S,t) of a set and a topology on it) is called a