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)