Reference

In a first-countable space, the relationship between sequential continuity and continuity is more straightforward. A topological space is first-countable if each point has a countable base of neighborhoods. This property allows us to use sequences to test continuity.

Key Points:

1. First-Countable Spaces:

   • A space is first-countable if every point has a countable local base (a countable collection of open sets where every neighborhood of the point contains one of these sets).
   • First-countable spaces include all metric spaces.

2. Sequential and General Continuity:

   • In a first-countable space, sequential continuity (where the limit of ( f(x_n) ) equals ( f(\lim x_n) ) for every sequence ( x_n ) converging to a point ( x )) implies general continuity.
   • Conversely, general continuity implies sequential continuity regardless of the space.

3. Importance of First-Countability:

   • The implication from sequential continuity to general continuity doesn't necessarily hold in spaces that are not first-countable.
   • In first-countable spaces, convergence can be checked using sequences, making the two concepts equivalent.

For more details, you can check this Math Stack Exchange post or other resources related to first-countability properties.