I’ve just begun studying about compact spaces and encountered a part which I don’t get.
In the note, it says (0,1] is not a compact space because {(1/k,1) : k=1,2,3,...} is an open cover of (0,1] with no finite subcover.
But I don’t see how the union of such intervals, which all have an open end at 1, can be closed at 1.
I’d be really grateful for any help. Thank you.
The given set is not a cover of $(0,1]$. There are two possible corrections. If “compact” is to be understood as “compact subset of $\mathbb R$”, then you can consider e.g. the open cover $\{(1/k, 1+1/k)\}$. If “compact” is to be understood as “compact as a topological space by itself” (which I think is the intended one), then the small fix would be to consider $\{(1/k, 1]\}$. The intervals $(1/k, 1]$ are open sets in $(0,1]$ when given the subspace topology.