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let alpha be a cardinal.

I want to look at topological spaces with the property that their topology has a basis with cardinality at most alpha. This property of a topological space certainly has a name. Could you tell how it is called?

For the case: alpha=aleph_0, this is just "second countable", but the property looks intersting even for bigger cardinals.

Thanks very much in advance

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A topological space $X$ is said to be of weight $\leq \alpha$ if it has a basis of cardinality $\leq \alpha$. So you're asking about spaces of weight at most $\alpha$.

The weight $w(X)$ of $X$ is the least cardinality of a basis. This is one of the most important cardinal characteristics of a topological space.

You can find a lot of basic information in Engelking's book on General topology and the standard references are books and articles by Juhasz, e.g. "Cardinal functions in topology" and the first article in the Handbook on set-theoretic topology.

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