A topological reflection principle equivalent to Shelah’s strong hypothesis

Abstract:

We notice that Shelah’s Strong Hypothesis is equivalent to the following reflection principle:

Suppose $\langle X,\tau \rangle$ is a first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $X$ is of cardinality at most $\kappa$, then the cardinality of $X$ is $\kappa$.