Definition talk:Compact Space/Topology/Subspace/Definition 2

Sutherland, edition 2 defines this as a "compact subset", and using open sets of X, not H. Edition 1 also uses open sets of X, and, strictly speaking, defines it for a subset while saying that he defines it for a subspace. But at this point in the book, Sutherland does not distinguish anymore between subsets and subspaces, so it must not be taken literally. --barto (talk) 10:36, 10 September 2017 (EDT)

The 1975 edition says:

With the above notation, $T$ is compact if given any cover $\mathcal U$ of $T$ by sets open in $S$, there is a finite subcover of $\mathcal U$ for $T$.

$T$ has been defined as:

a subspace of another topological space $S$.

This all matches our current definition. A subspace may be compact, a subset can only be compact if it has a topology imposed upon it, at which point it becomes a subspace. --prime mover (talk) 11:47, 10 September 2017 (EDT)

Indeed, because I just changed the definition so it would match. We're not using the topology of $H$ here. Take a look at the second edition. It corrects many mistakes from the first. --barto (talk) 11:51, 10 September 2017 (EDT)

I used to be really fit and strong, and I used to be able to tear telephone directories in half. Now I discover I can't even tear a copy of Sutherland in half. --prime mover (talk) 18:05, 10 September 2017 (EDT)