# A topological reflection principle equivalent to Shelah’s strong hypothesis

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

Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$.
If every separable subspace of $\mathbb X$ is of cardinality at most $\kappa$, then the cardinality of $\mathbb X$ is $\kappa$.

Suppose $\mathbb X$ is a countably tight space whose density is a regular cardinal, $\kappa$.
If every separable subspace of $\mathbb X$ is countable, then the cardinality of $\mathbb X$ is $\kappa$.