Compactness for "set" and "space"
I was wondering if there is any significance between the two settings. Do we treat them as two different things?
For example, let $(X,d)$ be a metric space with the topology induced by the metric $d$. From Heine-Borel theorem (for metric space), we know that the following are equivalent:
- Every sequence has a convergent subsequence;
- $X$ is a compact space.
And this is wrong about sets, take $\left\{\frac{1}{n}\right\}_n$ in $[0,1]$, it has a convergent subsequence, but it is only pre-compact (its closure is compact).
Edit: I guess if you take $\left\{\frac{1}{n}\right\}_n$ as a subspace, it does not have any convergent subsequence, therefore not a compact subspace (or set). So the two settings are actually the same?
The term space is often used for the set with structure, so compact set/space is somewhat equivalent. Yet, there may be some subtle difference in being formal. A topological space is a pair $(X,\tau)$ where $\tau$ is a topology on a set $X$, even though we often just write $X$ with $\tau$ omitted. You can say that a topological space is compact (since its topology is fixed), but you can't say that a set $X$ is compact since a set is not coming with a fixed topology. That's if you want to be formal, but in the daily math talks such subtleties may be redundant. What we often also say is that $A$ is a compact subset of $X$. To be formal, we have to say the subset $A$ of the set $X$ is compact in the subspace topology induced by $(X,\tau)$ but I think one rarely does say/write such a long sentence.
Regarding your example, by a convergence subsequence one means that the limit belongs to the same space. The limit of $\{1/n\}_{n\in \Bbb N}$ is $0$ which does not belong to this set.