Any space with a finite open cover.

Any space with finitely many points.

The real line.

Any infinite discrete space.

One open cover with no finite subcover.

Showing the space is infinite.

Showing some open cover is infinite.

Showing no open cover is finite.

$Y$ is compact.

$Y$ is a metric space.

$Y$ is finite.

Nothing, unless the map is one-to-one.

That $[a,b]$ is compact, and that a finite product of compact spaces is compact

That a closed subset of a compact space is compact, and that compact sets are bounded

That $\mathbb{R}$ is not compact, and that a finite space is compact

That continuous images of compact spaces are compact, and that $\mathbb{Z}$ is not compact

$C$ is open

$C$ is finite

$C$ is compact

$C=X$

Compact implies closed and bounded

Closed and bounded implies compact

Bounded implies compact

Closed implies compact

$\{(x,y):x^2+y^2<4\}$

$\{(x,y):y=x\}$

$\mathbb{Q}^2\cap[0,1]^2$

$\{(x,y):1\le x^2+y^2\le4\}$

It is finite

It is compact and complete

It is unbounded

Nothing without the least upper bound property

True

False