I am working from Brin and Stuck's textbook on Dynamical Systems. They quote the following: By compactness, there are finite (n,ϵ)-spanning sets. Can anybody explain why this is the case? I can sort of see that because of finite subcovers, the set must be finite, but why must there exist a spanning set in the first place?
They define (n,ϵ)-spanning sets as follows:
A subset A⊂ X is $(n, \epsilon)$-spanning if for every x ∈ X there is y ∈ A such that $d_n(x, y) < \epsilon$.
They define compactness by covers. For further context, see the following link to the passage I am working from (found online): http://wwwf.imperial.ac.uk/~jswlamb/m3a23/TopEntropyfromBS.pdf
Note: I asked a similar question previously, but cannot access to my account to update it.