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Does every first countable Lindelöf space have countable spread?

What's meaning of countable spread? see the definition of $s(X)$

$$s(X)=\sup\{|Y|:Y\subseteq X \text{ with the subspace topology is discrete}\}+\aleph_0.$$

Countable spread means that $s(X) = \aleph_0$.

Thanks ahead.

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up vote 4 down vote accepted

Let $X$ be the lexicographically ordered square; then $X$ is compact and first countable, but $s(X)=c(X)=2^\omega$.

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