ubcover).  The second axiom of
| countability is therefore definitely more restrictive than the first.
|
| It is worth noticing that, if U_1, U_2, ..., U_n, ... is a countable
| local base at x, then a new local base V_1, V_2, ..., V_n, ... can be
| found such that V_n contains V_n+1 for each n.  The construction is
| simple:  let V_n = |^|{U_k : k =&lt; n}.
|
| A 'subbase for the neighborhood system' of a point x, or a 'local subbase'
| at x, is a family of sets such that the family of all finite intersect