An alternative definition of compact sets

Question

I was wondering if the following alternative definition is the same (or weaker/stronger) than the default definition of compactness:

A set $K\subset \mathbb{R}$ is compact if every convergent sequence in $K$ converges to a point in $K$.

versus the usual:

A set $K\subset \mathbb{R}$ is compact if every sequence in $K$ has a subsequence that converges to a point in $K$.

It certainly applies to, say, $[a,b]$ and $(a,b)$ as well.

I've been searching for a case where it breaks down (i.e. one where the original definition applies but this one doesn't) but I was unsuccessful.

Answer 1

A set $K$ satisfies the first condition if and only if it is a closed subset of $\mathbb R$. For instance, $[0,\infty)$ is closed (and therefore satisfies the condition), but it is not compact.

The second condtion is equivalent to the compactness of $K$.

Answer 2

Take $X=\mathbb R$ for a counterexample. Every convergent sequence in $\mathbb R$ converges to a point in $\mathbb R$, however $\mathbb R$ is not compact.

If you want a proper subset, take $X=[0,\infty)$

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However, you are close. In fact, the following is true in all topological spaces:

A set $X$ is closed if every convergent sequence in $X$ converges to a point in $X$

(so, really, your condition is the condition for closedness, not compactness)

On the other hand,

A set $X$ is compact if every sequence in $X$ has a convergent subsequence.

which is slightly different.

Answer 3

Observe: a set $K \subseteq \mathbb R$ is closed $ \iff$ every convergent sequence in $K$ converges to a point in $K$.