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

Question

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!

Answer 1

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.