Mapping $U$ homeomorphically onto $f(U)$. I've got a rather elemental doubt about what "mapping a subspace homeomorphically onto its image" means, but I'd like to be sure what is being implied.
Let $f:X\to Y$ a continuous injective map between topological spaces. Let $U\subseteq X$ be an open subspace. Then we could say in some context that $f$ maps $U$ homeomorphically onto $f(U)$. 
My question is 

Must $f(U)$ be an open subspace of $Y$?

If we consider $f$ as a map $f:U\to f(U)$ (its restriction to $U$), then $f(U)$ is open in itself, so there is no contradiction with $f$ being a homeomorphism between $U$ and $f(U)$ if $f(U)$ wasn't open in $Y$. 

No, it doesn't have to be open in $Y$.
For instance, take $f\colon\{0\}\longrightarrow\mathbb R$, defined by $f(0)=0$. Consider in $\mathbb R$ the usual topology. Then $f$ maps $\{0\}$ homeomorphically onto its image, but $\{0\}$ is not an open subset of $\mathbb R$.
