Let $\kappa$ in an infinite cardinal and $X$ be a space and $e(x)=\kappa$, where the ``extent'' $e(X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. My question is this:

Under what condition, the space $X$ contains a closed discrete subset $Y$ such that $|Y|=\kappa$? It is known that if $X$ is metrizable, then we may find such subspace of $X$. What about if $X$ is a Moore space or other generalized metrizable spaces.

Thanks a lot.

Let $\kappa$ in an uncountable regular cardinal and $X$ be a space and $e(x)=\kappa$, where the ``extent'' $e(X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. My question is this:

Under what condition, the space $X$ contains a closed discrete subset $Y$ such that $|Y|=\kappa$? It is known that if $X$ is metrizable, then we may find such subspace of $X$. What about if $X$ is a Moore space or other generalized metrizable spaces.

Thanks a lot.