Is every continious image of a non-compact space is non-compact?

Question

Is every continuous image of a non-compact space is non-compact?

I was thinking about constant function. I think it will be false.

Am I right?

Answer 1

Let $f: [0, \infty) \to \mathbb R$ be defined by

$f(x)=1-x$ , if $0 \le x \le 1$ and $f(x)=0$, if $x>1$.

Then $f$ is continuous, $[0, \infty)$ is not compact, but $f([0, \infty))=[0,1]$ is compact.