Workshop on Set Theory and its Applications

These are the slides of a talk given at the Workshop on Set Theory and its Applications workshop (Weizmann Institute, February 19, 2007).

Talk Title: Nets of spaces having singular density

Abstract: The weight of a topological space X is the minimal cardinality of basis B for X. The density of X is the minimal cardinality of a dense subset of X. A net for X is a collection N of subsets of X such that any open set is the union of elements of N. Thus, any basis is a net. The theme of our talk is the following problem:

Find the least cardinal $\theta$ such that there exists a basis B for X, of cardinality equal to the weight of X, such that every open subset of X is the union of $<\theta$ many members of B.

Note, for example that if a space X is hereditarily Lindelöf, and B is a basis for X, then every open subset of X is the union of countably many members of B.

Thus, for a cardinal $\theta$, define the relative net-weight with respect to $\theta$ to be the minimal cardinality of a net N such that any open set is the union of $<\theta$ many elements of N.

The main result of this talk reads as follows:

Theorem. Assuming a very weak cardinal arithmetic hypothesis (PSH). If the density of X is a singular cardinal $\lambda$, then the relative net-weight of X with respect to $\text{cf}(\lambda)$ is strictly greater than $\lambda$.

In particular, in all currently known models of set theory, if X is a space of density and weight $\aleph_{\omega_1}$ , then X is not hereditarily Lindelöf.

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