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!



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


