Which statement is true for non-compact sets in a metric space

Question

We know that a set is compact if for every open cover, there exists a finite subcover.

If a set is not compact then is it true that:

1. There exists an open cover, such that there does not exists a finite subcover

2. For every open cover, there exists infinite subcover

3. For every open cover, every subcover is infinite

4. There exists an open cover, such that it has an infinite subcover

5. There are no open covers

Can someone illustrate with example?

Answer 1

If we know as an affirmative that a set is $X$ not compact, then

1. This statement is true. For if not, then it would mean that every open cover had a finite sub cover, concluding that the set indeed was compact, a contradiction.

2. $\{X\}$ is an open cover of $X$ and every sub cover of it is finite.

3. Same as above.

4. If $X$ had no infinite open covers then it would be a compact set. So take any infinite open cover and use it as itself as a sub cover.

5. $\{X\}$ is an open cover of $X$.