"If $f: X \to Y$ and $\dim(X) =\dim(Y)$, then $f(X)$ is open in $Y$" Is this a general fact?

Question

I am reading an article and I come to a statement that I am not sure if it is a general fact. In one line, the article says that $f:X \to Y$ where $\dim(X) = \dim(Y)$, so $f(X)$ is open in $Y$. Of course, there are some more properties in $f,X$ and $Y$ ($X,Y$ are open and connected, $f$ is differentiable). However, it seems to me that the author does not need any of them at all, because his statement is basically alone. I am wondering if this is a result in topology, which I am not very good at. So any help would be appreciated!

Answer 1

If $f$ is a constant function, then $f(X)$ is a singleton. And in general, singletons are not open sets.