📂Topology

Additive Compactness and Lindelöf Spaces

Definition ¹

1. If every countable open cover of $X$ has a finite subcover, then $X$ is called countably compact.
2. If every open cover of $X$ has a countable subcover, then $X$ is called Lindelöf.

Theorem

Countably Compact

• [1-1]: Every compact space is a countably compact space.
• [1-2]: Countable compactness is a topological property.

Lindelöf

• [2-1]: Every second-countable space is a Lindelöf space.
• [2-2]: If $X$ is Lindelöf, then $X$ being compact and $X$ being countably compact are equivalent to each other.

Explanation

Both cases merely add ‘countable’ to the concept of compactness, differing only in where it is added. Although Lindelöf is not frequently mentioned throughout topology, almost invariably, when it is mentioned, theorem [2-2] is used.

Being countably compact has its significance in being easier to prove than being compact.