# Is there a generalized order space $X$ with countable tightness which is not first countable?

I have a question concerning generalized order spaces.

Is there a generalized order space $$X$$ with countable tightness which is not first countable?

thanks a lot!

No, in general tightness and character are equal for GO-spaces: if $$x$$ is in the closure of $$(\gets,x)$$ then there is a subset $$L$$ of cardinality at most the tightness of $$(\gets,x)$$ that has $$x$$ in its closure; likewise if $$x\in\operatorname{cl}(x,\to)$$ there is $$R\subseteq(x,\to)$$ of cardinality at most the tightness with $$x\in\operatorname{cl}R$$. From these sets it is easy to make a local base at $$x$$ of cardinality at most the tightness.