Moreover, convergence to g(z) is uniform on compact subsets of the star.

The original definition above represents the special case where the direct system of compact subsets has a cofinal sequence.

The point at infinity should be thought of as lying outside every compact subset of X.

Note that, in a metric space, every compact subset is closed and bounded.

Therefore its image on the unit ball B must be a compact subset of the real line, and this proves the claim.

A compact subset of a Hausdorff space is closed.

The y-axis has infinite M-measure though all compact subsets of it have measure 0.

Let denote the set of all compact subsets of .

Let be a compact subset of and choose .

Any compact subset of R must be a countable set.