Examples

Because it’s always better with examples

Big classes of examples

Most commonly encountered topological spaces are paracompact, which makes it a pervasive notion.

• Obviously every compact topological space is paracompact;
• All metric spaces are paracompact;
• All regular Lindelöf spaces are paracompact. In particular, all CW complexes are paracompact;

One dimentional complex manifolds

Suppose we have a Hausdorff topological space with a holomorphic and one dimentional atlas, then surprisingly, it has to be paracompact [1]. Hence, there is no complex analogue of the long line.

Such a property is very specific to complex dimension one.

References

• [1] Ahlfors & Sario, Riemann surfaces