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It is well-known that the square principle □λ entails the existence of a non-reflecting stationary subset of λ+, whereas the weak square principle □λ∗ does not. Here we show that if μcf(λ) < λ for all μ < λ, then □λ∗ entails the existence of a non-reflecting stationary subset of Ecf(λ)λ+ in the forcing extension for adding a single Cohen subset of λ+.It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of □λ∗ for every singular cardinal λ of countable cofinality. [ABSTRACT FROM AUTHOR]

We shall introduce the notions of Λsp-closed and spg-closed sets. We also investigate properties of these sets and introduce some related new separation axioms. [ABSTRACT FROM AUTHOR]