If $X$ is a space, then the weak extent$we\left(X\right)$ of $X$ is the cardinal $min\{\alpha :$ If $𝒰$ is an open cover of $X$, then there exists $A\subseteq X$ such that $\left|A\right|=\alpha$ and $St(A,𝒰)=X\}$. In this note, we show that if $X$ is a normal space such that $\left|X\right|=𝔠$ and $we\left(X\right)=\omega$, then $X$ does not have a closed discrete subset of cardinality $𝔠$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $𝔠$, even if the condition that $we\left(X\right)=\omega$ is replaced by the stronger condition that $X$ is separable.