I have been trying to determine if the continuous image of a compact metric space is second countable.
I know that a compact metric space can be shown to be separable and that continuous functions preserve both the property of separability and compactness.
That being said to use Urysohn's theorem on would have to show that the image is metrizable to then use the proposed separability of the image to get that the image is in fact metrizable. From that point one could use Urysohn's to say that since the image is separable as well, it is second countable. So I don't know how to show, or if it is actually true that the image is metrizable.
The other thing to note is that if the map was open we would be done or if the image space was Hausdorff. That being said I know continuous images of Hausdorff spaces need not be Hausdorff.
I am not too sure how to proceed from here or if the statement is false (there's a counterexample). Any hint/help would be awesome.
The continuous image of a compact metric space need not even be first-countable.
Consider $X = [0,1]$ with the usual topology, and $Y = [0,1]$ with the co-finite topology. The identity function $f : X \to Y$ ($x \mapsto x$) is clearly a continuous function. However $Y$ is not first-countable, let alone second-countable.