Math Help - Compact Set (topology)

1. Compact Set (topology)

Let (X, d) be a metric space.

Show that any finite subset is compact.

2. Originally Posted by hebby
Let (X, d) be a metric space.

Show that any finite subset is compact.
Are you serious? Let $\mathcal{V}$ be an open cover for some finite subset $\mathcal{S}$ of $X$. There must exists some $O_s\in\mathcal{V}$ such that $s\in O_s$ for each $s\in\mathcal{S}$. Clearly then $\left\{O_s\right\}_{s\in\mathcal{S}}$ is a subcover of $\mathcal{S}$ and since $\mathcal{S}$ is finite so is $\left\{O_s\right\}_{s\in\mathcal{S}}$.

3. Hi

Thanks for the reply, could you explain what is this O small s?

4. These "O-sub-s" things are open sets; the union of these sets forms an open cover for your finite set.

Recall that one of the properties of a compact set is that every open cover of that set admits a finite sub-covering.

5. Originally Posted by hebby
Hi

Thanks for the reply, could you explain what is this O small s?
Originally Posted by h2osprey
These "O-sub-s" things are open sets; the union of these sets forms an open cover for your finite set.

Recall that one of the properties of a compact set is that every open cover of that set admits a finite sub-covering.
While , what h2osprey said was undeniably correct, I think the finer points got lost.

To be compact means that every open cover has a finite subcover. Thus, to show that a set is compact one way is to take an arbitrary open cover and derive a finite subcover. So, the concept is we let $\mathcal{V}$ be some open cover of $\mathcal{S}$. Next, we let $s\in\mathcal{S}$ be arbitrary, and since $\mathcal{V}$ covers $\mathcal{S}$ there exists some open set, which we will call $O_s$, in $\mathcal{V}$ such that it contains $s$. So, for each $s\in\mathcal{S}$ we take some corresponding $O_s\in\mathcal{V}$ and construct a new class $\left\{O_s\right\}_{s\in\mathcal{S}}$. This is clearly a subcover, and since there is at most $\text{card }\mathcal{S}$ elements of $\left\{O_s\right\}_{s\in\mathcal{S}}$ it must be a finite subcover.

The two finer points that you should definitely answer yourself is why can I choose this $O_s$ and why is there at most $\text{card }\mathcal{S}$ elements in $\left\{O_s\right\}_{s\in\mathcal{S}}$