On F-spaces

We present the class of $\text{F}$_t-spaces which is a subclass of the class of $\text{F}$-spaces of Harley and Stephenson containing many of the most interesting $\text{F}$-spaces, e.g. the Michael line, the Sorgenfrey line, Aleksandrov's double interval.

We prove that the $\text{F}$_t-spaces satisfy certain covering properties which $\text{F}$-spaces need not satisfy. In particular, (1) every neighborhood $\text{F}$_t-space is subparacompact, and (2) every $\text{F}$_t-space satisfies property L of Bacon. On the other hand, there are examples of neighborhood $\text{F}$-spaces which do not satisfy L.